f LIBRARY OF CONGRESS. 



.=m^.j/ 



•4: 







UNITED STATES OF AMERICA.^ 



Entered, according to Act of Congress, in the year 1869, by 

J. SHENTON ZANE, 

in the Clerk's Office of the District Court of the United States for the 
Eastern District of Pennsylvania. 

Stereotyped aud printed by King <fc Baird, 607 Sansom Str., Philadelphia. 



ELEMENTS OF ANALYSIS 



AS APPLIED TO 



THE MECHANICS 



OS 



ENGINEEEING AND MACHINERY, 



TREATISE 



THE MECHANICS 

OF 

Engineering and Machinery, 

WITH 

REQUISITE ANALYTICAL INSTRUCTIONS 

FOR THE UbE OF 

POLYTECHNIC INSTITUTES AND FOR THE REFERENCE OF 
ENGINEERS, ARCHITECTS, MACHINISTS, &c. 

By JULIUS WEISBACH, Ph.D., 

Councillor on Mines and Professor in the Eoyal School of Mines, Freiberg, Saxony; 

Knight of the Saxon Order of Merit, and of the Kussian Imperial Order 

OF St. Anne; Corresponding Member of the Imperial Academy of 

Sciences at St. Petersburg; Honorary Member of thk 

Society of German Engineers, &c. 



€xnn$htzb from llje J^otttt'^ '§,tbmh anb ^nlargeb ^txmmx €bitimt. 
By J. SHENTOIST ZANE, 

CIVIL AND mechanical ENGINEER, GRADUATE OF THE POLYTECHNIC COLLEGE AT CARLSR0HE 

(GRAND DUCHY OF BADEN). 

''0 INTHREEVOLUMES. 

VOL. I. 

"THEORETICAL MECHAlSriCS." 

Illustkated with 902 Engkavings. 




PHILADELPHIA : 
18 6 9. 



Entered, according to Act of Congress, in the year 1869, by 

J. SHENTON ZANE, 

in the Clerk's Office of tlie District Court of the United States for the 
Eastern District of Pennsylvania. 

Stereotyped and printed by King & Baibd, 607 Sansom Str., Philadelphia. 



'b 



t''^^ 



PREMCE TO THE SECOJ(D AMERICAN EDITION. 



Dr. Weisbach's treatise on tlie "Principles of Meclianics" 
is too well known to demand any explanation of its merits at 
the present time. The fact that it has been translated into at 
least four modern languages, is sufficient proof that it supplies 
a want long felt by scientific men, and a decided acknowledg- 
ment of its superiority over all other works of its kind. The 
practical engineer here finds not only the results of continued 
and profound study, but is also enabled, by the clearness and 
precision of the author's reasonings, to review each and every 
step leading to those results. 

The first American edition of the above work was issued in 
1848, and the fact of its having been a long time out of print, 
while the necessity for such a work is constantly increasing, 
is the only consideration which has prevailed upon the trans- 
lator to undertake a work requiring so much careful labor, 
and involving so great a responsibility. In the meantime, the 
original has reached its fourth edition, and has, by virtue of 
constant developments in the field of science, assumed much 
larger proportions. The prefaces of the author to the different 
editions are sufficiently explicit in regard to the nature and 
object of the original work, and as the translator has only 
sought an accurate reproduction of the same, no additional 
explanations are necessary. 

The only essential difference between the original and the 
translation is the occasional substitution of the English, for 
the Grerman and French, weights and measures. Yet, in all 
cases, the original co-efficients have been given, so that re- 
course may be had to them by the student, and a familiar 

(V) 



VI PREFACE TO THE SECOND AMERICAN EDITION. 

acquaintance with, tlie modes of foreign calculations thus main- 
tained. It is evident that readiness in transposing from foreign 
systems of computation to our own should be possessed by the 
practical engineer and machinist ; for foreign calculations are 
constantly copied into our scientific publications. 

Especial pains have been taken to retain the clear and accu-, 
rate style of the erudite author, and if the present edition 
shall occupy in our country the position which the work has 
taken in Europe, the most sincere wish of the translator will 
have been attained. 



AUTHOR'S PREFACE TO THE FIRST EDITION. 



It is not without some hesitation that I present the First 
Volume of mj elementary treatise on the " Mechanics of En- 
gineering and Machinery" to the public. Although I am con- 
scious of having composed this work with the greatest care 
and deliberation, I am, nevertheless, apprehensive of not hav- 
ing been able to satisfy the demands of all ; in fact, the views 
and requirements of different individuals are so various as to 
render my task peculiarly difficult. Some will find one or an- 
other chapter too minute, whilst others will find the very same 
too short ; some will require a more scientific treatment of cer- 
tain subjects, which others would have desired to have had 
presented in a more popular manner. But many years of study, 
much experience in teaching, and manifold observation, have 
indicated to me the method according to which I have pre- 
pared the present work, and which I have considered the most 
suitable for the intended purpose. 

My principle aim while preparing this work has been to at- 
tain the greatest simplicity in enunciation and proof, and to 
solve all important problems by means of the elementary mathe- 
matics only. When we consider the manifold knowledge to be 
acquired by engineers and machinists before they become cap- 
able in their departments, it is our duty. as instructors, to sim- 
plify in our. explanations the fundamental studies of science, by 
eschewing all superfluities, and by the application of the best 
known and most accessible auxiliaries. I have, therefore, in 
the present work, entirely avoided the application of the dif- 
ferential and integral calculus ; for although the facilities for 
learning these methods are no longer rare, it is, nevertheless, 
an unquestionable fact that without constant practice the nee- 

(VII) 



VIII AUTHOR S PREFACE TO THE FIRST EDITION. 

essaiy readiness in using them is veiy soon lost, and there are, 
therefore, many practical men, otherwise very efficient, who 
have forgotten how to apply them. As I am not of the same 
opinion as those authors who, in popular works, give the re- 
sults of the more difficult problems without proofs, I have pre- 
ferred to prove those which are practically important in an 
elementary, although sometimes prolix manner. Hence, in this 
work, a formula is seldom given without its derivation. Some 
general knowledge of certain laws of natural philosophy, and 
especially, a fundamental knowledge of pure elementary mathe- 
matics, is of course to be assumed in the study of this work. 
I have particularly endeavored to observe the just medium be- 
tween generalization and specialization; for although I am not 
igporant of the advantages of the former, I am, nevertheless, 
of the opinion that, in this, as in every elementary work, too 
much of it should be avoided. In practice, examples of a simple 
character occur miore frequently than complicated ones. It is 
also not to be denied that, in comprehensive examples, the 
simple and fundamental principles are frequently out of sight, 
and that it is often easier to derive the compound from the 
simple than the simple from the compound. 

This work should not be mistaken for a treatise on the con- 
struction of machines, as it is merely an introduction to the 
latter. Mechanics should hold the same relation to the science 
of the construction of machines, as descriptive geometry to the 
drawing of machines. After the knowledge of mechanics and 
descriptive geometry has been acquired, it seems most advan- 
tageous to combine the instruction on the construction and 
drawing of machines in one course. 

There may perhaps be a doubt of the advantage of dividing 
this work into theoretical and practical parts ; but if we con- 
sider that it is to impart instruction on all the mechanical rela- 
tions in the science of architecture and machinery, the utility, 
or rather the necessity, of this division becomes obvious. In 
order to be able to judge properly of any construction, partic- 



authoe's preface to the first edition. IX 

ularly of a machine, the most diverse knowledge of the laws 
of mechanics, as the laws of friction, strength, inertia, impact, 
efflux, &c., is requisite; therefore, the material for the compre- 
hension of architecture or machinery must he gathered from 
nearly all the departments of mechanics. IN'ow as it is in prac- 
tice much more advantageous to study the mechanical prin- 
ciples of each machine in connection than to be obliged to col- 
lect them from the various departments of mechanics, the 
utility of the adopted division seems beyond doubt. 

Having the practical application of principles constantly in 
view, I have sought, as far as possible, to illustrate the doc- 
trines advanced with appropriate examples, and I can also as- 
sert that, in the great number and appropriate selection of 
worked examples, this work excels many others of a similar 
nature. I also hope that the large number of carefully exe- 
cuted figures will be of great service in the study of this 
work. 

Especial attention has been devoted to the accuracy of the 
calculations, each example having been worked by three dif- 
ferent persons ; and it seems, therefore, hardly possible to dis- 
cover any considerable errors in the same. A careful inspection 
of the drawings will convince the student that they have been 
execut'ed with much care; in a subject of equal magnitudes, 
the dimensions of breadth or depth are, as a rule, made to 
appear only half as great as those of length and height. 

Finally, it is necessary to inform the reader that he will, in 
this work, find much that is new, and much that is peculiar 
to the author. Omitting many lesser articles which occur in 
almost every chapter, I will call the attention of the student 
to the more comprehensive subjects. A universal and easy 
method of finding the centre of gravity of plane surfaces and 
plane-surfaced polyhedra may be found at §§ 107, 112, and 113 ; 
an approximate formula for the catenary at § 148 ; and supple- 
ments to the theory of the friction of axes at §§ 167, 168, 169, 
172, and 173. The doctrine of impact has received essential 



X AUTHOR S PREFACE TO THE FIRST EDITION. 

additions at §§ 277 and 278, as the impact of inaperfectlj elas- 
tic bodies lias hitherto been too little regarded, and the case 
of a perfectly elastic body coming in contact with an imper- 
fectly elastic body, not at all. The greatest number of addi- 
tions, and some new laws, will be found in the department of 
hydraulics, the author having devoted himself to this branch 
for a number of years. The laws of the "Imperfect Contrac- 
tion of the Fluid Yein" first observed by the author, appear 
here for the first time in a treatise on mechanics. The most 
important results of the author's experiments on the efilux of 
water through slides, cocks, clacks, and valves, as also through 
short oblique, angular, curved, and long straight, tubes, are 
likewise given. The chapters on fiowing water, and the gaug- 
ing and impulse of water have also received some additions. 
The theory of the reaction of effluent water, as well as that 
of the impulse of water according to the principle of mechan- 
ical efiect, is entirely new. 

Although, upon the completion of the first volume, I some- 
times wish that I had treated some of the subjects in a differ- 
ent manner, I must, nevertheless, add that I have not yet been 
able to discover any essential defects. If some omissions are 

obvious, I must refer to the second volume, which will contain 

. *. 

supplements, intentionally reserved for it, as has been indicated 

in several passages of the present volume. 

My object in preparing this work has been to supply the 
practitioner with useful advice, the instructor with a guide in 
teaching, and the young engineer and machinist with a wel- 
come auxiliary in the acquirement of a knowledge of me- 
chanics ; and it will give me great satisfaction if my purpose 
has been attained. 

Freiberg, 3Iarch, 1846. 

JULIUS WEISBACH. 



AUTHOR'S PREFACE TO TUE SECOND EDITION. 



The second edition of tlie first volnme of tTie " MecTiamcs 
of Engineering and Macliinery" does not, in method and 
arrangement, essentially differ from the first edition. It is 
only by the development of the work, by the changes and 
additions which have been made, that the present edition has 
been considerably extended. The enlarged appearance of this 
edition is especially owing to three additions. The first con- 
sists of a concise and popular introduction of the infinitesimal 
calculus, at the beginning of the entire work. 

The object of this has been to avoid evolutions which are 
too complicated and elaborate by the lower calculus, and at 
the same time to render the student more independent in the 
important department of Mechanics. By the application of 
the rules contained in this introduction, it has been possible 
to take up many practically important subjects which could 
not be treated of, or at best, only incompletely, by elementary 
algebra and geometry. That those students who are not ac- 
quainted with the higher calculus may not be inconvenienced 
by its introduction, each paragraph in which it is applied, has 
been especially indicated by a parenthesis, as (§ 20). 

The second addition comprises a new chapter in hydro- 
statics, and treats of the molecular effects of water. As the 
knowledge of the molecular forces is of importance in hy- 
draulic and pneumatic observations and measurements, it has 
seemed to the author to be advisable to introduce the prin- 
cipal laws of these forces in a special chapter. 

Lastly, there is an Appendix treating of oscillations and 
undulatory motions, inasmuch as these are of importance, on 

(XI) 



XII AUTHOR S PREFACE TO THE SECOND EDITION. 

account of the influence they exert upon the motion, strength, 
and durability, of machines and other constructions. It is, 
moreover, to the observations of oscillations that we are in- 
debted for the most recent moduli of elasticity, which are of 
such very great practical importance. Magnetic force has also 
been considered in the Appendix, chiefly because of its great 
utility to the engineer in indicating his position in subter- 
ranean regions and places which afibrd no external view. The 
theory of waves, at the close of the volume, belongs wholly to 
the department of hydraulics; hence, no justification of its 
introduction in this work is necessary. 

The chapter on elasticity and strength has undergone ex- 
tensive changes, and received much additional matter; the 
department of hydraulics has also been much improved by 
continued experiments and corrections of the author. 

May this second edition also be the recipient of the con- 
sideration and approbation with which the first was welcomed, 
and which has encouraged the author in the further prepara- 
tion of the work. 

Freiberg, May^ 1850. 

JULIUS WEISBACH. 



AUTHOR'S PREFACE TO THE FOURTH EDITION. 



The fourtli edition of my "Mechanics of Engineering and 
Machinery," which I now submit to the public, has suffered 
no change either in the method or in the arrangement of the 
material. The somewhat rapid sale of three large editions of 
the work, and the appearance of two editions in the English 
language (in England and America), as also its translation 
into Swedish, Polish, and Russian, permit me to hope that 
this work now fully meets the exigencies of practical men, for 
'whom it is intended. Hence, in preparing this new edition, I 
have confined myself to removing observed errors, and to the 
incorporation of new, important, empirical results, and the- 
oretical developments. In the chapter on friction, for exam- 
pie, I have introduced the results of the latest experiments by 
Bochet, and the section which treats of elasticity and strength, 
has been revised 'with the aid of the most recent wri tineas of 
Lame, Rankine, Bresse, &c. The section on hydraulics has 
received manifold corrections and additions, especially, the 
results of recent investigations by the author. The experi- 
ments on the efflux of water under high and very high pres- 
sure, as also, those on the height of ascent of jets of water, 
and further, the comparative experiments on the impact of 
currents of air and of water, are of especial importance. The 
chapter on the efflux of air has been completely remodelled, 
the author being convinced that the general formulse for the 
efflux of air under high pressure do not correctly represent the 
law of efflux. The formulae obtained are very simple, as here, 
without diminishing the accuracy within tolerably extended 
limitSjWe have, in the known formula of heat 

(XIII) 



XIV AUTHOR S PREFACE TO THE FOURTH EDITION. 

l + ^^i _ fnY' 

1 + (5r Vy J 

changed tlie exponent 0,42 to 0,50, and thus put 

l + ^^^i |7i 



_ , (vid. § 461). 

1 -\- d T ^ Y 

The utility of a formula depends not so much upon its accu- 
racy for extreme limits, as upon its sufficient agreement with 
the results of experience, within certain limits. 

Several new paragraphs have been added to the paialytical 
laws of phoronomics and aerostatics, and in hydraulics the 
pressure of water flowing through tubes occupies two new 
paragraphs (439 and 440). In the chapter on the force and 
resistance of water, I have discussed the theory of the simple 
reaction wheel, and also its application in proving the theory 
of impulse and reaction of water. The more modern water 
and gas meters have also been treated of, as it is by the reac- 
tion of an effluent fluid that they assume a rotatory motion, 
the magnitude of which is easily estimated according to the 
foregoing theory. 

Lastly, the Appendix has received a small addition in refer- 
ence to Hagen's recent investigations on the subject of waves. 

If it be affirmed by some that the object of this book would 
have been more readily attained if it had received a more 
scientific treatment and been based upon the higher analysis, 
I must here reply that the work was written especially for 
the private study and reference of practical men, for whom, 
as a rule, the requisite knowledge of the diflerential and in- 
teo-ral calculus cannot be assumed. 

If, finally, my "Mechanics of Engineering and Machinery" 
has been repeatedly used in other works of the kind, I can, 
since literary honor is of much greater importance to me than 
pecuniary profit, only rejoice ; but in regard to writers who 
have appropriated portions of it without any acknowledg- 
ment, I can with safety appeal to the justice of public opinion. 

Freiberg, Maij, 1863. JULIUS WEISBACIL 



COl^TEKTS. 



Article Page 

1 — 4 Functions, Natural Laws 1 

5 — 6 Differentials, Position of Tangent 5 

7 — 8 Rules of Differentiation 7 

9—10 Tlie Function y^x"" , 10 

11—12 Straight Line, Ellipse, Hyperbola 15 

13—14 The Course of Curves, Maximum and Minimum 19 

15 Maclaurin's Tlieorem and the Binomial Series 23 

16 — 18 Integral, Integral Calculus 26 

19 — 23 Exponential and Logarithmic Functions 29 

24—27 Trigonometric and Circular Functions 35 

28 Reductions' Formula of Integral Calculus 40 

29—31 Quadrature of Curves 42 

32 Rectification of Curves 48 

33 — 34 Normal to, and Radius of Gyration of, Curves 50 

35 Function 2/ = ^ 55 

36 Method of the Least Squares 57 

37 Method of Interpolation - 60 



(XV) 



ELEMENTS OF ANALYSIS. 



Art. 1. The dependence of one magnitude y upon another x is 
expressed by a mathematical formula, as, for example, ^ = 3 ^'^, or 
y =^a ^™, &c. Generally, we put y =/(^), or 2; ^ c^ (7/), &c., and 
call y a function of ^, as also z a function of y. The signs /, ^, &c. 
merely signify generall}^ that y depends upon ^, or z upon y ; they 
leave the dependence of these magnitudes upon each other wholly 
undetermined, and do not, therefore, prescribe the algebraic process 
by which y proceeds from x^ or z from y. 

A function y=^f{x) is an indefinite equation; there are infinitely 
numerous values of x and y^ which correspond to the same, but if 
the one {x) be given, then the other (?/) is determined by the func- 
tion, and if the one be changed, the other, likewise, suffers a change. 
Hence, the indefinite magnitudes x and y are termed variables^ or 
variable magnitudes^ whilst those that are determined, or to be re- 
garded as determined, and, therefore, prescribe the algebraic process 
by which y proceeds from x^ are called constants^ or constant magni- 
tudes. That one of the variable magnitudes which is to be assumed 
arbitrarily, is termed the independent variable,, but that one which is 
determined as a function of the latter by a prescribed jjrocess, is 
called the dependent variable. In ?/ = a ^™, a and m are the con- 
stants, whilst X is the independent, and y the dependent, variable. 

The dependence of one magnitude z upon two others x and y is 
expressed by z =/(a7, y). In this case z is, at the same time, func- 
tion of X and ^, and here, therefore, we have to consider two inde- 
pendent variable magnitudes. 

Art. 2. Every dependence of one magnitude y upon another x^ 
expressed by a function, or formula y =/(^), may be represented 
by a plane curve, or curved line AP Q^ Eigs. 1 and 2; the abscissas 



Fig. 1. 



Fig. 2. 




A M N "^ ■" T M N ^ 

A Jf, A N^ &c., of the curve, correspond to the difi'erent values of 
the independent variable x, and the ordinates MP^ N Q, &c., to the 
1 (1) 



ELEMENTS OF ANALYSIS. 



[Art. 3. 




different values of the dependent variable y. The co-ordinates (ab- 
scissas and ordinates) of the curve, therefore, represent the two vari- 
ables of the function. 

The graphic illustration of a function, or the reduction of the 
same to a curve, combines many advantages: first, it aflbrds a sur- 
vey of the connection between two variable magnitudes ; secondly, 
it takes the place of a table of two combined values of a function ; 
and thirdly, it furnishes information in regard to the various prop- 
erties and relations of functions. The circle ADB^ Fig. 3, described 
by the radius GA= CB = r, which corresponds to the function 
Fig. 3. y ^=-]/ 2 r X — ^^, in which x and y represent the 

co-ordinates A l/and J/P, affords us, for example, 
not only a general survey of the different values 
which this function may assume, but also makes 
jB us acquainted with its other characteristics, since 
the properties of the circle have also their signifi- 
cance in the function. We comprehend from this, 
for instance, without further investigation, that y 
is zero, not only for x ^o^ but also for x = 2r; further, that y is a, 
maximum, and indeed = r, if ^ = r, &c. 

Art. 3. The laivs of natural pMlosopliy may, as a rule, be ex- 
pressed by functions between two or more magnitudes, and are, 
therefore, for the most part, capable of graphic illustration. 

1. For the free descent of bodies in a vacuum^ we have, for ex- 
ample, for the velocity of descent y which corresponds to the height 
of descent x^ y = -[/2 g x] this formula agrees also with the equation 
y = ■]/ p X of the parabola, if the parameter (p) of the latter be 
f)\it equal to twice the acceleration (2 g) of gravit}^ ; hence, the law 
of descent may also be graphically illustrated by a parabola AP Q^ 

Fig. 4, with the parameter p =:^2g. The ab- 
scissas A 31^ AN... of this curve are, of 
course, the spaces described in the descent, 
and the corresponding ordinates JXP, JSfQ... 
the corresponding velocities. 

2. If a be a certain volume of air under 
the pressure of one atmosphere, we have, ac- 
cording to the Law of Mariotte^ the volume of the same quantity of 

a 
air under a pressure of x atmospheres : y = — • 

a 



Fig. 4. 




for X 



For a? = 1, we have y =z a\ for ^ = 2, ?/ 
1^?2/ = T^; for ^ = 100, 2/ 



100 



for X 



for ^ = 4, 2/ : 

0; 



00, 



y 



Art. 3.] 



ELEMENTS OF A:JsrALTSIS. 



thus it is clear that the volume becomes less as the tension is greater, 
and that, if the Laio of 3Iariotte were accurate for all tensions, an infi- 
nitely small volume y would correspond to an infinitely great tension x. 
Further: ^= i gives 2/ = 2 a; ^ = i, ?/ = 4 a; 

hence, the less the tension the greater the volume, and if the tension 
be infinitely small, the volume will be infinitely great. 

The curve which corresponds to this law is represented in Figc 5, 
(&c.); J. Jf, AN . . . are the the tensions or abscissas x^ MP^ N Q ... 
the corresponding volumes or ordinates y. We perceive that this 
curve gradually approaches the co-ordinate axes A X and A F, with- 
out ever reaching them. 

3. The dependence of the expansive force y of vapor upon the 
temperature x ma}^ be expressed, at least within certain limits, by 
the formula 



fa -f- ^"A™ 
2/ = I — 7 — I atmosjoheres ; 



and we have from experiment within certain limits, a 

and ??i = 6. If, according to this, we put 

fib + x-\^ 



15,6 



Ho, 



Fig. 6. 




M 100 K 



and assume perfect accuracy for this formula, we obtain: 



For X = 100", y = T^Y = 



1,000 atmospheres, 



(t 



x= 50' 






X 



0\ 11 = 



y 
y = 



" x=~1b% y 



llOoJ 

vnsJ 



0,133 



0,006 



0,000 



ELEMENTS' OF ANALYSIS. 



[Art. 4. 



further, for x = 120", y = i - 



175 J 



« ^ = 200*', y = 



= 1,914 atmospheres, 



The corresponding curve is represented by P Q, Fig. 6 ; it passes 
through the axis of abcissas at a distance AO = — 75 from the 
origin A of the co-ordinates, and through the axis of ordinates, at a 
distance AS=^ 0,006 from this same point; further, the ordinate 
MP < 1 corresponds to the abscissa AM <^ 100, and the ordinate 
A^(^ > 1, to the abscissa AN^ 100; and it may also be observed, 
not only that y increases to infinity with ^, but also, that the curve 
ascends more and more perpendicularly the greater x becomes. 

Art. 4. A function z =f(x^ y) witli two independent variables 
may be illustrated by a curved surface B CD, Fig. 7, in which these 
variables x and y are represented by the abscissas A M and A N on 
the axes A X and A y, and the dependent variable z^ by the ordinate 
P of a point P in the surface ABC. If, with a determined value 
of x^ we give different values to ^, we obtain in z the ordinates of 
the points of a curve E P F running parallel with the plane of co- 
ordinates YZ ; but if, with a determined value of ?/, we assume 
different values for x^ there results in z the ordinates of the points 
of a curve G P H running parallel with the plane of co-ordinates XZ. 
Therefore, the entire curved surface BCD may be regarded as a 
continuous combination of curves running parallel with the planes 

of the co-ordinates. 

a{l~\-dy) 



The Law of Mariotte^ Gay^ and Lussac^ viz. z 



ac- 



X 



cording to which the volume z of a quantity of air may be calculated 
by the pressure x a.nd temperature y of the same, may be graphic- 
ally illustrated by the curved surface GKPH, Fig. 8. AM is the 
pressure x^ AN MO the temperature y^ and OP the corresponding 

Fig. 7. 



Fig. 8, 





Art. 5.] 



ELEMENTS OF ANALYSIS. 



volumQ z ; further, the co-ordinates of the curve P GH indicate the 
volumes for one and the same temperature AN^y, and the co-ordi- 
nates of the straight line KP^ the volumes for one and the same 
pressure AM^ x. 

Art. 5. If the independent variable of a function or abscissa 
AM=x^ Figs. 9 and 10, of the corresponding curve be allowed 
to increase by an infinitesimal magnitude MN^ which we shall in 
future designate by 9^, the corresponding dependent variable, or 
ordinate MP = y passes into NQ^y^^ and increases by the infini- 
tesimal value B Q = NQ — MP, which is to be designated by dy. 

Both increments dx and dy of x and y are called differentials, or 
elements of the variables or co-ordinates x and y, and it is now our 
chief task to find, for tlie most frequently recurring functions, the 
differentials, or rather, the relations between the associated elements 
of their variables x and y. If, in the function y=f(x), where x 
designates the abscissa A M, and y, the ordinate MP, we replace x 
hj x^dx = AM-^ MN= A N, we obtain, 

instead of ^Z, y -{-"cy = MP -|- B Q ^ NQ; therefore: 

y + dy=f(x -{-dx); 

and if, from this we abstract the first value of y, there will remain 
the differential of the variable y ; i. e.: 

'dy = df{x) =f(x + dx) -f{x). 



Fig. 9. 



Fig. 10. 




M N 



X A 




M N 



This is the most general rule for the determination of the differ- 
ential of a function, from which, by application to different functions, 
other rules more or less general may be deduced. 

If we have, for instance, y = ^^, then 

dy = (x-{-dxy — ^^, 
or, since we have to put 

(x -|- dxy = x^-\- 2xdx -J- ^^^ 
there will result 

dy = 2xdx + dx' = (2x + 8^) dx; 



c 



6 ELEMENTS OF ANALYSIS. [Art. 6. 

or more simply, since dx, as infinitesimal, vanishes when compared 
with 2 a;: 

dy = d{xy = 2xdoc, 

y=:x'^ corresponds to the area of a square AB G D^ Fig. 11, whose 

side AB ^^ AD is = ^; and we may infer from 

^^' ' the figure, that, by the augmentation of the sides 

N PQ lojBM=DJSf=dx^ the square is enlarged by 

_i j_Q two rectangles BO and DF ^ 2xdT. and a 

square OF = (dxY; that, therefore, with an in- 
finitesimal increment (dx) of x^ the square y = x'^ 
increases by the element 2xdx. 
^~M Art. 6. The straight line TF Q, Figs. 9 and 

10, which passes through two infinitely approx- 
imate points P, Q^ of a curve, is called a tangent, or line of contact 
of this curve, and determines the direction of the same between these 
points. The direction of the tangent is indicated by the angle 
F TM=a under which the axis AX of abscissas is intersected by 
this line. If the curve be concave, as AF Q, Fig. 9, the tangent will 
lie outside the curve and axis of abscissas ; but if the curve be 
convex, as AF Q, Fig. 10, it mil lie between the curve and axis of 
abscissas. 

In the infinitesimal right-angled triangle FQR, Figs. 9 and 10, 
having the catheti F F = dx and F Q = dy, the angle QF F is equal 
to the tangential angle F TM= «, and, since we have 

tang. QFF==^^, 
we have also : 

tang.a = ^^; 

therefore, the quotient of the two differentials dy and dx indicates 
the trigonometric tangent of the tangential angle. 

For the parabola, for instance, w^hose equation is ^/^ =px, we have, 
if we put y"^ ==px = z, 

dz = (y-j- dyf — y' = y' + 2ijdy + dy' — y' = ^y'dy + dy\ 
or, since 8^/^ vanishes when compared with 2ydy: 

dz = 2ydy, 
and likewise: 

dz =p(x -f- dx) — pdx. 

There results, accordingly, 2ydy =pdx, and hence, for the tan- 
gential angle of the parabola : 

dy p 2/^ ^ 

tang, a ^ ^~ = -^ = -^— = --. 
'^ dx 2y 2xy 2y • 



Art. T.J 



ELEMENTS OF ANALYSIS. 



Generall}?-, we term the determined portion P T of the line of con- 
tact between the point of contact P and the point of intersection T 

Fig. 13. 
Pig. 12. 




with the axis of abscissas, the tangent^ and its projection TM in 
the axis of abscissas, the suhtaiigent^ and we thus have 
subtang, = P 31 cotang. P TM 

= y cotang, a = y 



for example, for the parabola: 

suhtang. 



dy 



y 



— == 2 ^. 

y 

Here, therefore, the subtangent is equal to twice the abscissa, and 
the position of the tangent for every point P of the parabola is, 
accordingly, easily indicated. 

For a curved surface BCD Fig. t, the angles of inclination a 
and /5 of the tangents P T and P C/' at a point P are determined by 
the formulae: 

tanq. a = — - and tana. (3 = ^-. 

ox cy 

The plane P TV laid through P T and P CT' is a tangential plane 
of the curved surface. 

Art. *!. For a function y =i a -{- ^mf (^), we have: 

dy = \_a ^ mf {x -|- 8^?)] — [«- + m/ {x)\ 
■=i a — a -\- mf {x -{- dx) — mf (x) 
= m [/ (x J^dx) —f {x)2 ; i. e.: 

L . . . . . d [a -\- mf (x)2 = 7ndf (x) ] 
e.g.: 

8 (5 4- 3 ^^) = 3 [(x + dxy — ^2] = 3 . 2 xdx = Qxdx. 
There is likewise : 

d(i — ^x')= — ^d (xf = — ^[(x-^ dxy —x'2 

= — ^ (x^ -{- S x"^ dx ^ 3 xdx^ -\- dx^ — x^) 

1 Q /v*^ /n 'Y* *^ 'Y*~ /I ^ 

. "o" • O tXy \J tX/ - "o" tXy %y tAy 

We may, accordingly, adduce the following important rule : The 
constant memhers (a, 5) of a function vanish by differentiation^ and 
the constant factors (m, 3) remain thereby unchanged. 



8 



ELEMENTS OF ANALYSIS. 



[Art. 8. 



The accuracy of this rule may also be graphically illustrated. For 
the curve AF Q, Fig. 14, whose co-ordinates are first, AM=a; and 

Fig. 14. Fig. 15. 





MF =y =f(x)^ and ailerwards, A^M^ = x and M^F^=a-\-y = a 
-j-/(^), we have F B = dx' and F Q =^dy = df(x), and also = 8 
(a -\~ y) = d[a 4-/(^)] 5 and Tor the curves AF^ Q^ and AF Q^ Fig. 
15, whose associated ordinates MF^ and 31 F^ ^Q-^ and NQ have a 
certain relation to each other, the relation between the differentials 

F^Q^ = NQ^ — MF^ and FQ = NQ — MF is always the same; 
for if we put 31 F^ = m . 31 F^ and N Q^=^m.N Q, there follows 

B^Q^ = NQ^ — 3IF^ = m{NQ — M:F)=m.QR; 

i. e.: 

8 [m/(^)] = m df(x). 

If we have, further, y = u -\~ v^ therefore, the sum of tivo variables 
u and iJ, there is also 

8?/ rr= 16 -[- 8it -j- -u -j- 8 u — (u -\- v) ) \. Q.^ from Art. 5 : 

II "^(u -{- v) = du -{-dv ; likewise: 

8 [/ W + <P (^)] ='df{x) + 8 ^ (x). 
Thus, the differential of the sum of several functions is equal to 
the sum of the differentials of the single functions ; e. g. : 

d(2x-\- Sx^'^^x^) = 2dx^Qxdx-~^x'dx = (2^Qx — ix')dx. 

The accuracy of this rule is also verified in the curve AF Q, Fig. 
15. If we have MF ■=f{x) and F F^ = (p (^), there is also 
MF^ = y ^f{x) -\- (p{x), and 
dy = R^Q^ = B^S-^SQ, = RQ-\-SQ, = df{x) + 8cr W; 
as we may put F^ S parallel to F Q, and hence B^S =^ B Q and 
QS = FF^. 

Art. 8. If we have y = uv^ therefore, the product of tivo vari- 
ables^ as the area of a rectangle AB G D^ Fig. 16, with the variable 
sides AB =^u and B G = v^ there results 

dy = (u -\- du) (v -\- dv) — uv = uv -\- udv -\- vdu ~\- dudv — uv 

=zudv -\- vdu -\- dudv = udv -}- (v -\- dv)u. 



Art. 8.] 



ELEMENTS OF ANALYSIS. 



Fig. 10. 




B M 



But in D -|- 8u, dv is infinitesimal in compar- 
ison with V ; hence we may put 

V -\- dv = v, and (v -[- d'v)du = vdu, 
as also, 

udv -f (d -f- dv)du=^udv -j- vdu; 
so that there follows : 



IIL 



d(uv) = udv -\- vdu, 



and 

8[/(^) . c.(.^)] ^f(x)d<p(x) + <p(^)df(x). 

Therefore, the differential of the product of two variables is equal 
the sum of the products of the one, and the differential of the other, 
variable. 

If the sides of the rectangle AB G D, Fig. 16, increase 'bj BM= 
du and D = dv, the area y =:^ AB .AD = uv of the same wiU in- 
crease by the rectangles C ==udv, C 31= vdu, and GP = dudv, 
the last of which, beinrr infinitely small in comparison with the others, 
vanishes ; hence, the differential of this area is only to be put equal 
to the sum udv -{- vdu of the areas of the two rectangles G 
and GM. 

According to this rule, we have, for instance, for y=zx(Sx'^'j- 1) : 

dy = xd ,3 ^2 _|_ 1) ^ (^3 ^2 _|_ 1^) 0^ ^ 3 ^g (^.^2^) ^ ^3 ^2 _^ X) dx 

= 3^.2 xdx -\- ^ x'^dx ^dx = {^ x' ^ 1) dx. 
There is further, ifw designate a third variable factor: 
8 (uvw) = ud (uw) -\- vwdu; or, 
since we have d (ow) = vdw -j- wdv: 

d (uvw) = uvdw -[- uwdv -\- vwdu- likewise, 
d (uvwz) = uvwdz -\- uvzdw ^ uwzdv -j- vwzdu. 
If we-have u = v = w = z, there follows d (u^) = 4:U^du, as also 
gjer -rally: 

•'r/. ..... d (x'^)= m.x'^-^dx, 

if the exponent m be a positive whole number. 

For example: 8 (^^) = 1 x^dx, and 8 (|^0 = ^ x^dx. 
If, in y = x~'^, m be again a positive whole number, we have also 
yx"^ =^1, and 8 {yx^) = 0; i. e.: 
yd {x^) -f x'^dy ^= 0, and hence, 



dy = 
or, if we put - 



ydix"^) 



X" 



. mx 



dx 



x^ 



mx~ 



-^dx; 



x"^ 
— m = n: 
d{x'') = nx:^~'^dx. 
. Therefore, rule lY. applies also to powers with negative whole 
numbers for exponents. For example: 

38^ 



8 {x-^) = — ^x~'dx 



X 



4 1 



10 ELEMENTS OF ANALYSIS. [Art. 9. 

likewise : 

8 (3 ^' + 1)-' = - 2 (3 x^ + 1)-^ 8 (3 xO = - (3^f_f ""ly 

m 

If in 1/ = x"^. — be a fraction whose denominator n and numera- 

^ ^ n 

tor m are wliole numbers, we have also y^ = ^™, and 8-(^") = d (^"'); 

i. e.: ny^~'^ ^2/ = m^™~-^ dx; hence, 

„ m x'^~^ dx m x^~'^ dx m --i 

<^?/ = — ^^=r- = — ^_m = ^ ^"^ 8^- 

liC n 

772/ 

If we put — = p, there follows 

n 

dy = d (x^) ^^^P~^ dx, which likewise corresponds to rule lY., 
now generally regarded as accurate. 

We have also d (u'') = p wp~^ du, if u designate some dependent 
function of x. 

From this we have, for instance: 

S (y'^) = d (xh) = ^xV^ dx = i V^dx, 

J 8 (2r^ — ^'■-) 2r8^ — 2^8a7 (r — x) dx 



^'^ 21/ 1^ t/ 2 



rx- — X' 



u 
To find the differential of a quotient y = — , let us put u = vy, 

from which du = vdy ^ ydv; consequently, there follows 

cu — ycv V 

■ = J 



dy = ^^ ^:i:i_ : ; 1. e.; 



V 

vdu — udv 



V 

We have, therefore, for example: 
fx' — 1~\ (x -{-2)d (x' — 1) — (x' _ 1)8 (^ -f- 2) 

__ (a7 + 2) . 2^8^ — ( x' — 1) . 8^ _ r^Mif^+j-'A o 



-f 2J - (^ + 2)^ 

/r8^ — (x"^ — 1) . 8^ ^ fa 

-(^+"2)^ - I" (^ + 2)^ 



and further, 

Art. 9. The function y = x'"- is the most important of the entire 
analysis, as it occurs in almost every investigation. If, to the expo- 
nent n we ascribe every possible value, positive and negative, entire 
and fractional, &c., it will also produce the most diverse curves, 
as may be seen in Fig. 11. Here, A is zero, or the origin of the co- 
ordinate axes X X and Y Y. 



Art. 9.] ELEMENTS OF ANALYSIS. H 

If, at either side of the co-ordinate axes, at the distances x=±l 
and y == ± 1 from A, we draw the lines X^ X^, X^ X^, Y^ F^, and 
Y^ T^ parallel to these axes, and connect their points of intersection 
P , P , P , and P, by the transversals ZZ] Z, ZT, we obtain a dia- 
gram with which the curves corresponding to the equation y = x"" 
are more or less connected. Moreover, we have for every point of 
the axis of abscissas XXj ?/ = 0, as also for every point of the axis 
of ordinates Z TJ ^ = ; further, for the points in the axes X^ X, 
and X^ X^, 2/ = ±: Ij and for those in the axes Z, Y, and Y^ Y^, 
^ = zh 1. 

If, in the equation y = x""^ we put a; = 1, we always obtain ^/ ^ 1, 
no matter what kind of a number the exponent n may be, and onl}^ 
for certain infrequent values of ?2, can y ^ — 1. Consequentl}^, all 
the curves proper to the equation y ^ x"" pass through the point P^ 
having the co-ordinates AM=1 and AN =\. 

If we assume n = 1, and thus make ?/ = ^, we obtain the 
straight line ZAY^ deviating equally from the two axes XX and ZZ, 

which ascends on the one side of A at an angle of 45° (^"t- Ji ^^^ 

descends at the same angle on the other side. On the contrary, for 
?/ = — ^, we obtain the straight line Z^ A'Z[^ descending at an angle 
of 45*^ on the one side of A^ and ascending at the same angle on the 
other. 

If, on the other hand, we have n > 1, y = x"" must be less than 
X for ^ < 1, and greater than a; for ^ > 1 ; and if we have n < 1, 
y = x'' must be greater than ^ for ^ < 1, and less than ^ for ^ > 1. 

To the first case (71 > 1) correspond convex curves^ which, at the 
beginning, run below the straight line ZAZ^ but which, from P^, 
run above the same, v.^hilst to the second case {n <^ 1) correspond 
concave curves in which the reverse occurs. 

If, in the first case, the exponent n be assumed as becoming less 
and less and finally vanishing, or approximating zero, the ordinates 
will approximate more and more to the constant value y = x^ = 1, 
and the corresponding curves above AX^ to the broken line ANF-^X^; 
but if, in the second case, the exponent 7i become greater and greater, 
the ordinates will gradually aj^proxim^ate to the limiting value y = x'^ 
= SCO = 00 , whilst the abscissas will gradually approximate to the 
limit X =y^ ^1] and hence, the corresponding curves wiU approach 
nearer and nearer to the broken line A MP^ Z^. 



12 



ELEMENTS OF ANALYSIS. 



[Art. 9. 



If we assume n-= — 1, tlius putting y = x ^ = — , we shall have 



for X 



; and we have therefore 



0, ^ = 00, and for x = oo^ y 

to consider a curve 1 P^ 1 (treated of in Art. 3, and illustrated in 
Fig. 5,) which always approaches the axis of ordinates on the one 
side, and the axis of abscissas on the other, but without ever reach- 
ing either the one or the other. 

Fig. 17. 




If the exponent ( — n) of the function y =z x ^ = — -^ be a proper 

1 1 

fraction, we shall have for ^ <^ 1, y <C — <> and for ^ ^ 1, ly ^ — ; 

but if this exponent be greater than unity, we shall have for x <^ 1^ 
?/>> — , and for x ^ 1, y <C - • Therefore, the curves correspond- 
ing to the function y = x"^ run, at the beginning, below or above, 
and afterwards, from the point P, above or below the curve y = x~'^ 

= -- -, according as n is less or greater than unity. Whilst generally 
the curves corresponding to the positive values of n run first below, 



[Art. 9. ELEMENTS OF ANALYSIS. 13 

and ft'om P^^ above, the straight line X^ X^. those proceeding from 
negative exponents ( — n)\ pass first above, and from P^, below, the 
line X^ IX^. In the former curves, we have for ^ = 0, also y = 0^ 
and for ^ = oo, also y = cc ] whilst in the latter, we have for ^ = 0, 
^ = oo, and for ^ = go, ?/ = 0. If the former depart more and more 
from the co-ordinate axes XlK and Y Z, the farther we follow them 
from the origin J., the latter approach more and more the axis XX 

on the one side, and the axis Y^ on the other, but without ever 
reaching them. 

Moreover, the last systems of curves approach nearer and nearer 
the broken line YNP^X^^ or the broken line Y^P^MX, according as 
the exponent approximates the limit n =^ or n = oc. 

If, in y =z X ' m be an odd ivJiole number (1, 3, 5, Y . . »)■, y 
has the same sign as x; also, positive values of y correspond to 
positive values of ^, and negative values of i/, to negative values of j7. 
If, on the other hand, m be an even whole number ('2, 4, 6 . . . ), 
y will be positive for both positive and negative x. The curves in 
the first case (as 3 P^ AP^ 3, or 1 P^^ 1, 1 Pg 1) run, consequently, on 
the one side of the axis of ordinates, above, and on the other, below, 

the axis of abscissas XAX] in the second case, however, (as 2 P^ 

J- P^ 2, or 2 Pj 2, 2 P^ 2) the curves run only above the axis of 
abscissas, and occupy, therefore, only the first and fourth squares. 
The former, for m = ± oo, correspond to the limiting lines Y^ MAM^ Y^ 
and X M Y^^ XM^ Y^^ whilst the latter correspond to the limiting lines 
r, MA M^ Y^ and XMY^, X 31^ ^2- 

li.my = x "^ , 71 be an odd whole number, y has the same sign 
as X', and if n be an even whole number, every positive x will give 
two equal values for y, one positive, and one negative, whilst for 
every negative x, y is imaginary or impossible. The curves (as -^ P^ 
A P3 ^) which correspond to the first case, are, therefore, only to be 
found in the first and third squares, and those for the second case 
(as i P^ AP^^), only in the first and second; the former have, for 
m-= 00, the lines of limitation X^ NAN~X, and X^ NY, X^N^ Z; 
the latter, the lines X, JSfA X^ X^ and X^ N F, X, N^Y 

As y = X " requires x = y , it follows that the last system 

of curves \^y = x'''j does not deviate from the foregoing \y = x 7 , 
except in its position in regard to the intersection of the axes, and 



14 ELEMENTS OE ANALYSIS. [Art. 10. 

that, by turning the curves, those of the one system may be brought 
to coincide with those of the other. 

^ f ly 1 

Since we have y = x"^ = yx"^ j = (^™)", the general course of the 

corresponding curve may always be given from the foregoing. Eor 

example, the curve for 

y = x^ = {x^y = (f-y, 

has positive ordinates both for positive and for negative x. On the 
otlier hand, the curve for 

y = xi= (X^^y = (]/^)% 
has positive ordinates for positiA^e x only, and the two are, indeed, 
opposite. Further, in the curve for 

y has always the same sign as x^ as neither the fifth root nor the 
cube changes the sign of the base. 

m 

Lastly, the curves corresponding to the equation y = — x""^ differ 

m 

from those of the equation y = x^ ^ only in their opposite positions 

in reference to the axis of abscissas XX^ and constitute the symme- 
trical halves of a whole. 

Art. 10. From the important formula d {oo"^) = nx^~^dx there 
follows also the formula for the tangential angle of the corresponding 
curves, illustrated in Fig. 18 ; we have, namely: 

'dy — 

tana, a = ^-- = nx""^ \ 

- ^ dx ' 

and hence, the subtangent of these curves : 

dx x^ X 

dy n x''~^ n ' 
There is, accordingly, for NeiVs parabola, whose equation is 

Vx^ 
a • . 



1 ^(^') „ 1^ .a_3J^ 
a 



tang, a = _ ^' >^' ^ = --^. | x'^ = ^ 

V a 

and the subtanofent = ^ x. 



V'a 8^ V~' ' "^ "^ "' 



a? _ 

Further, we have already for the above curve y =^ ■ — ■= a^ x ^ : 



tang, a = a 



,d(x-') _ a' ___ fay 

^ JU ^ 



dx X 



X 

and the subtangent = — z- = — x, (Comp. Fig. 5.) 



Art. 11.] 



ELEMENTS OF ANALYSIS. 



15 



Consequently, we shall have 

for ^ = 0, tang, a =. — oo, therefore, a == 90"; 
further, for x = a^ tang, a = — 1, therefore, a = 135°; 

and for x ==co^ fang, a = 0, therefore, a = 0^] &c. 



Fisc. 18. 




Art. 11. If a straight line AO^ Fig. 19, intersects the axis of 
abscissas under the angle OAX= «, and is distant from the origin 
G of the co-ordinates by GK = n, the equation between the co-or- 
dinates GM= JSfP == ^ and GN= 31 F = y of Si point F in the 
line, will be, since we have n = MB — ML and MR = y cos. a, as 
also ML = X sin. «, 



X sin. a 



y cos a 
For X = 0, y assumes the value G B ^:=h 



n. 



n 



-; hence, we have 

h -\- X 



COS. a 
also n = b-cos. a, and y cos. a — x sin. a = b cos. a, or y 
tang. a. 

The lines G A and GBhy which the points of intersection A and B 
of the straight line with the co-ordinate axes G X and G Y, are 



16 



ELEMENTS OF ANALYSIS. 



[Art. 11. 




a 



distant from the origin (7, are gene- 
rally termed the parameters of the 
straight line, and are designated by 
the letters a and h. According to the 
figure, we have G A = — a, hence: 

GB b 

tang. « = -^^ = - -^, 

and consequently, the equation of the 
straio-ht line: y = b — — ^, or: 

1; (vid. Ingenieur. page 164). 



When a curve approaches nearer and nearer, ad infinitum^ to a 
straight line which is distant by a finite magnitude from the origin 
of the co-ordinates, without ever reaching the same, this straight 
line is called the asymptote of the curve. 

The asymptote may be regarded as the tangent or line of contact 
for an infinitely distant point of the curve. Its angle of inclination a 
to the axis of abscissas is, therefore, determined by 

tana, a = ---. 

and its distance n from the zero of the co-ordinates, by the equation 

n = y COS. a — X sin. a = (y — x tang, a) cos. a 
y — X tang. 



s-('-4^■^l■+(s■. 



, ■ y 1 -\- (tang, a) 

as also by n =^ (y cotg. a — x) sin. a = 



y cotg. a — x 
V 1 -{- {cotg. ay 



if we put X and y = oo. 

In order that a tangent for an infinitely distant point of contact 
may be an asymptote, it is necessary, that, for ^or y = oo, y — x 
tang, a, or y cotg. a — ^, be not infinitely great. 



For a curve of the equation y = x~ 



— , we have 



tang, a 



m 



— - T-r and y 



X' 



X tang, a = x~^ -j- 



77^ 
x^ 



in -\- 1 



x^ 



as also y cotg. a — x ■=^ 



X 

m 



X 



X 



— (m -I- 1) — ; hence: 
^ ni 



1. for ^ = GO, ?/ = 0, tang, a 

and 

2. for ?/ = 00, ^ = 0, tang, a 



0, 2/ — X tang, a 
00, y cotg. — X 



0, and n = Q] 



0, and n 



0. 



Art. 12.] 



ELEMENTS OF xiNALYSIS. 



IT 



But the axis of abscissas XX corresponds to tlie conditions a := 
and n = 0, and the axis of ordinates FY, to the conditions a = co 
and n = ()] hence, these axes are, at the same time, asymptotes of 
the curves wliich correspond to the equation y = x~^, (Comp. tlie 
curves 1 Pj, 2P^2, and JP,|, in Eig. 18, page 15.) 

Art. 12. The equation of an ellipse ADA^D^, Fig. 20, may be 
immediately deduced from the equation 



X' 




Vi 



of the circle ABA^B^^ having 
the radii GA=GB= CP = a, 
and the co-ordinates GM=x 
and MP = ?/^, if it be taken 
into consideration that the or- 
dinate MQ = y of the ellipse 
stands in the same relation to 
the ordinate MP = y^ of the 
circle (the abscissas being the 
same), as the minor semi-axis 
C P> = b of the ellipse to the 
radius CB ^= a of the circle. 
We have, therefore. 



y 



a 



-— = — ; hence y^ = — y, and x' -]- -— y' = a'-, 



2/i 
i. e 



a 



X 



2 y2 

-j- Y^ = 1, as the equation of the ellipse. 



b- for -j- b\ we obtam the 



a^ ■ b 
If, in this equation, we substitute 

equation 

x"^ y"^ 
-T — lY =^ 

of the hyperbola consisting of two branches P A Q and Pj A^ Q^, 
Fior. 21. 



If, in the formula 



y = — V 



a 



x"- 



thus obtained, we take x infinitely great, a^ wiU vanish in comparison 
with x\ and 



y = -i/^' 



bx 
± — = ^ X tang, a 



wiU be the equation of two straight lines C U and C V passing 
through the origin C of the co-ordinates. As the ordinates 

±: — x = — -/ ^2 and — / ^2 _ ^2 



a 



a 



18 



ELEMENTS OF ANALYSIS. 



[Art. 12. 



approach more and more to equality, the greater we assume x to be, 
it follows that the straight lines C U and C V are the asymptotes of 
the hyperbola. 

If we take G A = a^ as also the perpendiculars AB = -\- h and 
AD =:^ — 6, we can thereby determine the two asymptotes; for we 
have, for the angles ± a under which the axis of abscissas is inter- 
sected by the asymptotes: 

tavg. AG B = -j^A i. e.: tang, a = — , and likewise: 




-W ■ T 

If the asymptotes U U and F F be taken as co-ordinate axes, if, 
further, the abscissa or co-ordinate CA^in the one axal direction be 
pnt = w, and the ordinate or co-ordinate NP in the other, = u, 
there will result, since the direction of u deviates from the axis of 
abscissas CXby the angle a, and that of u, from the same axis by 
the angle — a, the abscissa: 

G M = X = G N COS. a -\- NP cos. a = (u -\- v) cos. a, 
and the ordinate: 

MP ^ y = G N sin. a — NP sin. a = (u — v) sin. a. 

If we further designate the hypothenuse GB = V a^ -\- b'^ by e, 

we have: 

a ^ . b 

COS. a z=z — and sm. a = — : 



,, COS. a sin. a 1 ^ 

consequently: = • — ; — = — and 

^ *^ a be 



X' 



y"^ {y}-^^uv^rr) 



COS. a^ 



(u^ — ^uv -f- v"^) 



w^ -f- 2wu -f- -u^ 



u' 



2wu -\- v"^ 



b' 

4:UV 



sm. a^ 



ELEMENTS OF ANALYSIS. 



19 



Art. 13.] 

from which there results the so-called equation of asymptotes of 
the hyperbola: 

u V 



e"" e" 

-—, or -u = — . 

It is, therefore, easy to draw the hyperbola between the given asymp- 

totes. The co-ordinates for the vertex A are GE = EA =-^, whilst 

Fig. 22. 

-T '^ 




those for the point K are C B ^r= e and BK 



— ; we have, further, 



-, i — , &C. 



for the abscissas 2e, 3e, 4e, &c, the ordinates J -.-, ^ 

Art. 13. If, in the elementary ratio -^-, or in the formula for the 

tangent (tang. «, Art. 6,) of the tangential angle, we substitute suc- 
cessively different values for ^, there will result the different posi- 
tions of the line of contact of the appurtenant curve. If we take 
X = 0^ we obtain the tangent of the tangential angle at the origin 
of the co-ordinates; but if we take ^ = oo, we have the tangent for 
an infinitely distant point of the curve. The points where the tan- 
gent to a curve runs parallel with one of the axes of the co-ordinates 
are the most important ; because, here, as a rule, the one or the other 
of the co-ordinates x and y has its maximum or viinimum value ; 
or is, as we sa}^, a maximum or a minimum. For the parallelism 
with the axis of abscissas, we have a = 0; therefore, also, tang. a:=:0; 
and for the parallelism with the axis of ordinates, a =: 90*^; therefore, 
tang, a = oo; and from this follows the rule: The values of the 
abscissa or independent variable x^ to ivhich the maximum or mini- 
mum values of the ordinate or dependent variable y correspond, 

dy 
mo,y be found by pidting the differential ratio --- = and = oc, 

ex 

and resolving the equations obtained, ivith respect to x. 



20 



ELEMENTS OF ANALYSIS. 



[Art. 14. 



For the equation 2/ = 6 ^ — ^ x^ -j- x^^ whicli corresponds to the 
curve AF QE in Eig. 23, we have, for example, 

^ " dx -}- Sx' = 3 {2 — 3x ^ x') = 3 (1 — x) (2 — x)] 



= 6 



ox 



dy 



and if we put pr- = 0, there results: 

ex 

1 — X = and 2 



X 



0, 



i. e.: X = 1 and x 

Fig. 23. 
Y 



2. 



If these values be put in the formula y = Qx 
■ — 1^^ -h ^^ there will result the maximum value 
of y: 31 P = 6 — |H- l=f, and the minimum 
value: ^(^ == 12 — 18 + 8 = 2. 

We have, further, for the curve KOPQB 
Fig. 24, whose equation is 

g^ = to«<? « = 1 + |(^ _ 1)-* = 1 + ^^j-- ; 




and, indeed, = for 



— - J =_ 1; i. e. for ^ J/: 

3F ^ — 1 



^ 



= 1 - {%y 
1. 



— ^9 = 0, tost; on the other hand, = 00 for ^i\^= ^ 
To the first case corresponds the maximum value: 

MP = y^ = l- (§)3 + (1)^ = U = 1,148; 
and to the last, the minimum value: 2sfQ=='-y^ = ^' 

Fig. 24. 

There is also, further, for 
X =^ 0^ A = y ^= 1 ] on the 
other hand, 2/ = for the ab- 
scissa AK= x^ which corre- 
sponds to the cubic equation 
x^ -\- x"^ — 2 ^ -(- 1, and has 
the value x = — 2, 148. 

Art. 14. In a curve as- 
cending from the origin A^ y 
-2 _1 A MN +% increases with ^, and dy is 

therefore positive, whilst in a descending curve^ y decreases as x 
increases, and dy is, therefore, negative, and finally, zero, at the 
point where the curve runs parallel with the co-ordinate axis AX] 
likewise, the elements of ordinates : 

S Q = P S tang. QP S^ i. e. dy^ = dx . tang, a^, 
TP = QT tang. R Q T, i.Q.dy^ = dx . tang, a^, &c., 
corresponding to like elements of abscissas dx = MN = NO = 
PS= QT , , . 




ELEMENTS OF ANALYSIS. 



21 



Art. 14.] 

are, in a convex curve APJR, Fig. 25, on the point of increase, whilst 
in a concave curve APR^ Fig. 26, they are on the point of decrease. 
The same is also true of the tangential angles a^, a^? &c. Conse- 
quently, there is, in the first case: 

d (tang, a) = 8 I — I positive, 

and in the second: 

d (tang, a) = d i^ ) negative. 



Fis. 25. 





Fig. 27. 




A M JST O "^ A M N O " A M N O 

Lastly, we have for the point of inflection Q, Fig. 2t, i. e. for that 
point of the curve where convexity passes into concavit}^, or vice 
versa, S Q 



Tit, and hence: 



m - »• 



d{^ 



d (tang, a) 

Therefore, the following rule is applicable: If the differential of 
the tangent of the tangential angle be positive, the curve will be con- 
vex, if it be negative, the curve will be concave, and if it be zero, 
we shall have to consider a point of inflection of the curve. 

From the above, it is also easy to infer the following: The point 
where the curve runs parallel with the axis of abscissas, for which 
there is, therefore, tang, a := 0, corresponds either to a minimum, 
or maximum, or to a turning point of the curve, according as this 
curve is convex, concave, or neither one or the other \ i. e. according 
as we have: 

6 (tang, a) positive, negative, or zej''o. 

On the other hand, the point where a curve runs ]3arallel with the 
axis of ordinates, for which we have tang. a= cc, corresponds to a 
minimum, maximum, or to a turning point of the curve, accordiug 
as the same is concave, convex, or partly concave smd partly convex ; 
therefore, according as 8 (tang, a) is negative or positive before and 
after this point, or has a sign before it different from that which 
follows. 

A portion of the curve with point of inflection Q, of the first kind, 
is shown in Fig. 28; and one of the second kind, in Fig. 29. We 
see the corresponding ordinate NQ is neither a maximum nor a 



22 



ELEMENTS OF ANALYSIS. 



[Art. 14. 



miinmum; for in no case are the ordinates MP and OR simulta- 
neously greater or less than N Q. 

Li geometry, physics, mechanics, &c., the finding of maximum 
and minimum values of a function is often of great importance. As, 
in the sequel, it will often be necessary to determine such values of 
functions, only the following geometrical problem will here be solved. 



Fig. 28. 



Fig. 29. 






M N O AM N^O" 

Eequired, the dimensions of a right circular cylinder AN^ Fig. 30, 
which has, with a given content F, the smallest surface 0. If the 
diameter of the base of this cylinder be designated b}^ ^, and the 
height of the same by y^ we have; 

TT 



V 



X y, 



and the surface, or the area of the two bases plus the area of the 

envelope; 

2-^2 

= —^ + ^ xy, 

or, since, according to the first equation, we may put 

4 V 

t: y = ■ — —, therefore, tz x y = 4: V x"^: 

and consequently, as we can treat and x as co-ordinates of a curve : 
tana. a. = — — = -k x — 4 V x^"^. 

If, now, we make this quotient zero, we obtain the equation of 

condition: 

4 V 

-K X ■=^ — — , or 77 ^^ = 4 F, 



the solution of which leads to 



X 



JAZ, and 



y = 



4 F 



TT X^ 



3/ 64 V2 _j^_ _ 'M 



V 



= X. 



Since, further, d (tang, a) ==: i tt -\- —j- j dx is positive^ tliis result 



leads to the minimum souu'ht. 



Art. 15.] ELEMENTS OF ANALYSIS. 23 

This result is applicable also, when it is required to find the 
dimensions of a cj^lindrical vessel, which requires, with a given 
capacit.y, the least amount of material. It corresponds directly 
to this case, if the cover of the vessel is to be like the circular 
bottom; but if no cover is required, we have 

--=^ -|- 4 F^"~^, consequently: 

T. X 4c V 
■ = , from which there follows: 

2 x'' ' 



^ = 2 ^— , and y = ^ ~ , y^ = ^— = ^ x. 

Whilst, therefore, in the first case, the height of the cylinder is to 
be taken equal to its width, in the second, the height is to be taken 
only equal to one half of its width. 

Art. 15. By successive differentiation of a function y =f(x), 
we find an entire series of new functions of the independent variable 

dy df(x) 



fA^) 



dx d X 



For y =f (x) = x'^, for example, there follows: 

/i (^) = I ^KA (^) = V ^~KA (^) = — U ^~s &c- 

For a function which is represented in a convergent series pro- 
gressing according to the powers of x, whose exponents are pos- 
itive whole numbers, as 

?/ =/ W = ^0 + A ^ + A ^' + A ^' + A ^* H 

we obtain 

f(x) = A^-i-2A.^x + 3A^x''-}-AA^x'-\ 

/, (^) = 2 ^, -h 2 . 3 ^3 ^ + 3 . 4 J^ ^^ H 

/3 (^) = 2 . 3 ^3 + 2 . 3 . 4 ^^ ^^ H &c. 

If, in these series, we make ^ = 0, we obtain expressions which 
niaj be used for finding the constant co-efficients A^, A^, ^2 • • • ^'^^- • 

/ (0) = A, A (0) = 1 A, A (0) = 2 J„ /3 (0) = 2 . 3 . A„ &c., 

and hence, the co-efficients themselves: 

A =/(0), A =/: (0), A = iA (0), A = ^A (0), 
A = 27^ /. (0), &c. 

Accordingly, a function may be developed into the following 
form, termed Maclaurin^s series: 



24 ELEMENTS OF ANALYSIS. [Art. 15. 



/W =/(0) +/, (0) • T +/2 (0) • o +-^^ ("^ ■ 1:2:3 

+^'(o)-i:i^4 + -" 

For the binomial function y =^ f {x) = (1 -\- xY^ we. have 
f, {x) = n (1 + xY~\f {x) =n{n — l){l-\- xY~\ 
f {x) =n(n — 1) (n _ 2) (1 -f xy-% &c.; 
therefore, if we put ^ = 0, there results: 

/ (0) = 1,/, (0) = «,/, (0) = « ( » - 1), 
/3 (0) = n (» -!)(«_ 2), &0., 
and there follows the binomial series: 



I. (1 + ^)'^ = 1 + -- ^ -|- 



n n (7? — 1) 



1 . 2 



x"^ 



"We have further: 

/-I Nn -. '^ , 71 (n — 1) , nfn — 1) (^ — 2) , , 

(l = xy=l — —x-}- \ ^^ ^ ^' — 1 , 2 73 + '" 

as also: 

(1 + x)-- = 1 _ — ^ + ^^ ^^ ^ x' 1 . 2.3 "^ '■* 

1 X 

If we put 1 -j- ^ = (1 — z)~^ = ^, there follows z= , 

X — — Z X ~j— tX? 

and 

(1 + xy = (1 — z)-" = 1 + mz + " *■" "^ -^^ z^ 

n (n + 1) (n + 2) '. 

+ TTTT^ ' +---i.e.. 

n(72 + l)(n + 2) /- ^ Y 
^ 1.2.3 VI -f ^J ~f" ** 

The series under I. is a finite one for entire positive values of n, 
and that under 11. is a finite one for entire negative values of n. 
For example, 

(1 + ^)5 = 1 -[- 5 ^ 4- 10 ^2 _|_ 10 ^^ + 5 a;* + x^, and 
Since we have a -^ x = a ll -\ u there follows also: 



[Art. 15. ELEMENTS OF ANALYSIS. 25 

III. (a -^ xy = a^ + ^ a--' x + ^^f ""/^ a'^-^ ^^ 

n (n — 1) (n — '2) 



1.2.3 



For example: f 1009^ = (1000 + 9)t = 100 (1 + 0,009)3 

= 100 (l + I . 0,009 + '^ ^^ ~ ^^ . (0,009)^ -1 ^ 

= 100 (1 + 0,006 — 0,000009) = 100,5991. 
There is also: 

(^ -f 1)^ = ^^ 4- Ti^'^-i H V ^ ^^-2 -j 

hence, for very great values of x^ approximately: 

{x -\- ly = ^"^ -j- nx""-^. 

f^ _i_ x)"^ x"^ 

Accordingly, there follows x^~^ = ; further: 

(^ _ l)--i == '^'^ 

(x — ^y-"- = 



x"- 


-{X- 


— 


i)» 






n 




? 




(X- 


-ly 




(X- 


- 2)"^ 






n 






(^,- 


-2y 




(x — 


■ 3)'^ 



n 



2"" 1"" 

and lastly: 1^-^ = . 

Ey adding to both sides of the equation, there follows now: 

^u-i _[. (^ _ 1)11-1 -\- (x— 2y-'- H- (^ — sy-"- -1 \- 1 

__(x -\- ly — I'' 

~ n ' 

or if we put n — 1 = m, therefore, n :=: m -f 1, and invert the series : 

1« + 2» + 3» + . . . + (37 _ 1)» + ^» = ^ W^^ • 

Further, since x should be infinitely great, we may put (x -j- 1)™+^ 
__ ^m+i . ^i^ence there follows the sum of the poioers of the natural 
progression of nnmhers: 

lY. 1°^ -f 2"^ + 3"^ H [- x"^ = -: for example: 

m -\- 1 

if 1^ + 1^2^ + if ^ + if T^ + h if 1000^5 approximately 

5 



1000^^ 

3 



= I ^1000^ = 60000. 



26 



ELEMENTS OF ANALYSIS. 



Art. 16.] 





E^ 




---^ 




rv"^ 




j 




c/ 




K 






B 






II 








/ 




a 











r L M N 



o 



Art. 16. The ordinate OF =y, corresponding to the abscissa 
^^^' ^^' AO = x^ Fig. 31, ma}^ be composed 

of an infinite number of unequal ele- 
ments 8?/, as FB, G C, HD, KE . . . 
which correspond to the equal elements 
dx= AF = FL = LM = MN ... of 
the abscissa. If, therefore, we had given 
'dy^= (f {x) .dx^y would be found by 
summation of all those values of dy^ 
which result, when, in ^ (x) . dx, we 
substitute for x successively §j7, 2dx^ odx^ 4dx . . . up to ndx = x. 
This summation is indicated by the integral sign f w^hich is placed 
before the general expression for the elements which are to be sum- 
med up; thus, instead of 

y = l<p {dx) + cp {^dx) + cp (38^) + . . . + s^ {x)'\ dx, 
we write y = /V (x) dx. 

In this case y is called the integral of (p {x) dx^ as ^Iso (p (^x) 
dx, the differential of y. 

Oftentimes, the integral y^ (x) dx may be determined by actual 

summation of the series <p (8^), (f (28^), ^(38^')^ ^^'j i^ i®, however, 
much more simple to make use of one of the rules of the integral 
calculus^ which are to be developed in the sequel. 

If n be the number of the elements (8^) of .2;, therefore, x = ndx^ 

X 

ov dx = ■ — , we may put: 
n 

For the diff'erential dy^axdx, we have, for instance, the integral; 

y =J'axdx = adx (dx -\- 2dx -f- 38^ + \- ndx) 

= (1 + 24-3 + h ^^) a'^^'j 

or, since, according to Art. 15, lY., we have for ?i = 00, the sum of 
the natural progression of numbers l-]-2-f3-j- ... -\- n = ^ n\ 

and dx^ = 



71" 



i n^ a 



X' 



n' 



■q" t/' Jb m 



y = fa xdx 

In a similar manner we find: 

y=J^{x)dx=^^ = \idx^ 



(28j:)'+(38j7)^-[-...-f028x)^ 



8^ 



-= (1 + 2^ -1- 3'^ + 



a 



if we put .37 = n8^, or assume it to consist of n elements dx. 



Art. n.] ELEMENTS OF ANALYSIS. 21 

But according to § 15, lY., we have, for n = oo, 

9?" 

1 + 2- -f 3^ H 1- n^ = — ; hence there follows: 

3 



/ 



'X'dx n^ dx^ (ndory x^ 



3 a 3 a 3a 

Art. it. From the formula 8 [a -f- 7?i/(j?)] = mdf{x)^ we have, 
by inversion, fmdf{x) = a -f mf (x) = a -{- mfdf{x)] cr, if 
we put 8/ (^) = cff (a;) . dx: 

I. Z"??! c? (^) dx =^ a -\- iiifcp (^x) dx] 
and hence it follows: that the constant factor m remains unchanged 
b}' integration as well as b}^ differentiation; that an existing constant 
member a cannot be determined by integration alone; and that, 
therefore, mere integration produces an integral which is still unde- 
termined. 

To find the constant member, two associated values of x and y = 
f ^ (^) 'ox must be known. If we have for x ^ c^y =^k,, and if we 
liave found that y =^ T (p (^x) dx = a -{- f{x)^ there must be also: 

h = a -i- f{c), 
and the subtraction gives: y — k ==■ f {x) — f (c)', therefore, in this 
case: 

y =fe {X) -dx = k+f{x) -/(c) =f{x) + k -f(c); 
and we have, accordingly, the constant a = k — f (c). 

If it be known, for example, that the undetermined integral 

/x- 
xdx =^ — gives y = ^ for ^ ^ 1, we have the req[uired 

constant a = 3 — i = |; and hence, the integral: 

r X'- 5 -\- X' 

y =Jxdx = a ^ -^ = ■ — ^ — . 

Even the determination of the constant, leaves the integral still 
undetermined ; since, for x as independent variable, any arbitrary 
value may be assumed ; but if we require an entirely det«erminate 
value k^ of the integral, which corresponds to a determined value c^ 
of x^ the latter must be, further, introduced in the integral already 
found; therefore k^=^k -f /(<^i) — /(^)* 

Thus, for X = 5^ y = i xdx ^ — - — gives y = 15. 

That value of x for which y = 0, is the best known; in this case, 
therefore, we have ^ ^ 0, and hence the indeterminate integral 
f<p (x) dx=f (x) leads to the determinate integral k^=f (c,) — (c), 
which is found, if, in the expression / (x) for the former, the two 
given limiting values c^ and c of ^ be introduced, and the ascer- 
tained A'alues be subtracted the one from the other. 



28 ELEMENTS OF ANALYSIS. [Art. 18. 

To indicate this, we write, instead of i <p (x) 8 ^, I <p(x)dx; if,tliere- 

fore, we have j ^ (^) 8^ = — -, J (p(^x)dx = -^—— — . 

The inversion of the differential formula 8[/ (^) -f cp (ir;)] := 8/ 
{x) + S^ (^), gives the integral formula: J\jdf {x) -[- 8^ (^)] =/ 
(^) + S^ (^); or, if we put 8/(^) = v'' W 8^, and 8c^ (^) =/ (^) 8^: 
IL /[</' (^) 8^ + / {x) 8^] =/v^' (^) 8^ +// {x) 8.r. 

Therefore, ^/^e integral of a sum of several differentials is equal 
to the sum of the integrals of the single differentials. 

For example, f(fi -\- bx) dx =^f^ dx -\- f^ ocdx =i 2>x -\- f ^^. 

Art. 18. The most important differential formula TV. of Art. 8: 

8 (^°) = nx^~'^ 8^, 
leads, by inversion, to the equally important integral formula. There 
is, accordingly, /*n a? "^""^ dx = x^^ or nfx'^~^ dx = x^^ hence: 

x"" 



j 



x"^ '^ dx ^= , 

71 



if, therefore, we put n — 1 == m, and consequently, n = m -j- 1, 
we obtain the following important integral: 



/. 



X^ dx :=^ 



m-f-1' 
which occurs in practice as often, at least, as all the others together. 

The form of this integral indicates also that it corresponds to the 
system of curves treated of in Art. 9, and illustrated in Fig. 17. 

From this we have, for example, 

fbx^dx=^ b J x^ dx = ^ x^] further, 

j i^x^ dx = I xi dx = ^ xs = ^ -^ x'''j 
rdx , r _i^ , xi — 

y(4 — Qx'-\-b x') dx =f4: dx — /6 x' dx +/5 x' dx 
= 4:fdx — ^fx} dx 4- hfx^ dx = 4: X — 2 x^ ^ x'-, 
and, if we introduce 3 x — 2 = w, therefore, 3 8^ = 8«, or 8^ 

du 



jy^ Sx — 2.dx =fu—, 



1 du , u^ 



o 3 3 y |/ 



= ll/(3^-2y; 
lastly, if we put 2 x'' — 1 =: w, therefore, 4 x dx = du, or xdx 

_du^ 



Art. 19.] ELEMENTS OF ANALYSIS. 29 

15 ,/_ 15 . 

By appending the limiting values, the indeterminate integrals are 
immediately changed to determinate ones ; for example, 

f\x'dx = ^^ (2*_P) ={ . (16 — l) = 18f, 

'9 dx 



X 



2 Vx 

f 



1/9 —1/4 =h 



K l/3^_2 . 8^ = 1 (i/16^ — 1/10 = 1(64 — 1) = 14. 

If, for instance, we had for x = 0,/(4 — 6^' + 5 ^0 8^ = T, 
we should have generally: 

y (4 — 6 x' ^ ^ x') dx = 1 -i- 4: X — 2 x^^ -^ x\ 

Art. 19. The so-called exponential function y = a^^ which con- 
sists of a power with variable exponents, maj^, by aid of 3Iaclaurin^s 
Theorem^ be developed into a series, in which process, the corre- 
sponding differential will also be found. 

If we put a"^ = Aq -^ A^ X -]- A.^ x^ -{- A^ x^ -\- . . . , or, (since for 
^ = 0, a"^ assumes the value a'^ = 1^ and we have, therefore, A^ = 1,) 
a"^ = 1 -\- A^ X -\- A^ x"^ -{- A^ x^ -{- . . . , we have also : 
a^^ = l -^ A^ dx + A^ dx^ -{- A^dx^ -^ . . ., and hence: 
d (a^) = a'^+^^ _- a^ = a'^ a^^ — a"" = a"" (a^^ — 1) 
= a- (A^ dx + ^2 dx'\ A^ dx' -{- . . .) 
= a"" (A^ -}- A.^dx -^ . . .) dx = A^ a^'dx. 
Now, from successive differentiation of the series, there follows: 
f (x) = a^ = l -j- A^x -{- A^x' -i- A^x' -\ , 

•^^ ^""^ ^ ^?' = A<^'' = A + ^A^-\-^A^' + "-^ 
A W = ^^fp = ^.^ a- = 2 ^, + 2 . 3 . ^3 ^ + . . . , 

' /aW^^fi^^—^:^ ^^ = 2.3.^3 + ..., 

hence, if we put ^ == 0, there results: 

A^ = J^, 2 A^ = A,\ 2 . 3 . A.^ = A^' -{- • " , whence, 

^= = r^ ^- ' ^= = TTTTz ^' ' ^' = 1 . 2 ! 3 ■ 4 -^' "^^•' 

and the exponential series takes the form: 

4- A' 1 

^ '1.2.3.4^ 



30 ELEMENTS OF ANALYSIS. [Art. 20. 

The constant co-efficient A-^ is, of course, a determinate function 
of the constant base, as also, the hitter is a function of the former; 
hence, if one of the two numbers be given, the other is thereby 
determined. The simplest, or so-called natural^ series of poioers^ 
is obtained for A^^l^ whose base (a) is, in the sequel, to be indi- 
cated by e. There is, therefore, 

II. e^ = 1 + -- + -_+ — -_ + ,_,_3., + • • •. 

and if we put ^ = 1, there will result the base of the natural series 
of powers: 

e^ = e = 1 + 1 + i + i + 2^ + • • • -= 2,n82828 . . . 

1 
If we put e = a"', or a = e^, we have ~ = nat. log. a, the so-called 

natural^ or hyperholic logarithm of a, and 

IIL a- = (eiy = e^ = 1 + ^(-""A + -^- f-Y 

^ 1 . 2 . 3 ^mJ ~^ *' 
Since this series coincides in form with that under L, we have 

also A, = ■ — , and 

^ m 



a^ doG 



lY. 8 {a^) = A^ a^ '^ X ^= = nat. log. a . a^ 8 57, as also: 



m 



Y. 8 (e^) ^e^'dx. 
For example: 8 (e'^+0 = e'"^' 8 (3 ^ + 1) = 3 6^^+^ dx. 

If we put y = a^ = e"^, we have inversely: 

SG = log.^y and — = nat. log. ?/, hence: 

log.^ y = m nat. log. 2/, and inversely, 

nat. log. ?/, or log.^ y ^= — • ^^9-^ V- 

The number m is called the modulus of the system of logarithms 
corresponding to the base a. Therefore, by help of the same, the 
natural logarithm may be changed into any artificial logarithm, and 
vice versa. For Briggs^ system of logarithms, the base is a = 10, 
hence -^ = nat. log. 10 =: 2,30258 . . . , and inversely, the modulus: 

m = — —- = 0,43429 . . ., 

nat. log. 10 

therefore, 

log. y r= 0,43429 nat. log. ?/, and 
no,t. log. y = 2,30258 log. y. 
Art. 20. The course of the curves whicli correspond to the expo- 
nential functions y =^ e"" and ?/ = 10"", is illustrated in Fig. 32. 



Art. 20.] 



ELEMENTS OF AT^ALYSIS. 



31 



For ^ == 0, Tre have in both cases y^=e^^=a^=l\ hence, also, both 
cnrves OQS and OQ^S^ pass throngh the same point {0) m the 
axis of orclinates AY. For ^ := 1, we have 

y = e^ = 2,tl8 . . . and 

2/ = 10^ = 10, 
for X = 2, 

y = e^ = 2,n8^ = T,389 and 

y = 10^ = 10^ = 100, &c.; 
therefore, on the positive side of the axis of abscissas, both cnrves 
ascend very perpendicnlarly ; and particularly the last. On the other 
hand we have, for x =^ — 1 : * 

- = 0.363 . . . and 




^ 2,n8 . 

= 10-1 = 0,1; 



2; 



0,135 and 10^ 



further, for x = • 

pX p — 1 

^ -^ ~2,n8^ 

= 10-2 = 0,01; and, lastl}^, for 
X =^ ■ — 00, both equations give; 

e-" = — = — = 0. 

Consequently, on the negative side 
of the axis of abscissas, the two 
curves approach nearer and nearer 
to the same, the latter, indeed, more 
abruptly than the former, but an 
actual coincidence with the axis 
never takes place. 

As from y = e^, there results 
X. = nat. log. ?/, and likewise from 



y 



X = log. ?/, these curves 



furnish a scale of the natural loga- 
rithms, and of those of Briggs ; the 
abscissas are, ^dz., the logarithms 
of the ordinates, and we have, for 
example, 

AM= nat. log. MP 
= log.^ MP^, &c. 
According to the differential for- 

2 — 1 T AM 1 2^ — ^ mula lY. of the last article, the 

tangential angle a of the exponential curve is determined by the 
simple formula: 

6?y a^ dx a^ y 



tang, a = ^^ = - — 



cx 



m cx 



111 



on 



= y nat. log. a. 



32 ELEMENTS OE ANALYSIS. [Art. 22. 

In the curve OP, Q^S^, Fig. 32, the suUangent is, consequently, 
^=ycotg. a := m, therefore, constant; and in the curve OF QS, it is 
always = 1 ; for example, for the point Q, ~A1 = 1 ; for the point i?, 
12 = 1, &c. 

Art. 21. If there be ^ = a^, there is further: 

dx = d (ar) = .?!l^, 
m 

and inversely: 

r, m d X md X 

^ ay X 

But we have also y = log.,^ x] i. e. the logarithm of the variable 

power X with the constant base a, whence there results the following 

differential formulae of the logarithmic functions 

y = log.^ X and y = nat. log. x: 

T r^ /7 . md X 1 dx , 

I. c (log. X) = ■ = . — , as also, 

X nat. log. ax 

f)x 
II. d {nat. log. x) = — . 

If a be the tangential angle of the curve which corresponds to the 

equation y = log.^ x, we have also tang, a = ■ — , and the suhtangent^ 

X 

Xlj 

^ycotg.a=-^, 

therefore, proportional to the area xy of sl rectangle to be con- 
structed from the sides x and y. 

By means of the differential formulae I. and II. we obtain: 

, o/ .7 V--^ 8tX^ d{xi) x-^dx dx 

1. d (nat. log. V x) = — — r- = - — ^ = i i — = o-, 

yx x^ x^ ^-^ 

or also = 3 (^ nat. log. x) = ^ d (nat. log. x) = ^ 



X 



(2 -I- x'\ 
-^- ) = 8 [log. (2 + ^) — log. x'^ 



= 8 log. (2 + ^) — 8 log. (x') 

dx 2 ^- — — ^^ "^ "^^ ^'^ 



^-\- x' X X (2 -{- x) 

8. 8 (nat. log. ^^-^-j) = ^ h«^- %• (e^— 1)1—8 [nat. log. (e^+l)] 
8 (e'O 8 C^'') _ g"" 8^ _ g"" 8-^ _ ^ e^'d X 

Art. 22. If the differential formulae of the foregoing article be- 
come inverted, we meet with other important integral formulae as 
follows. 



[Art. 23. ELEMENTS OF A:N^ALYSIS. 33 

From 8 (a^') = , there follows | 

^ ^ 777. J 



= a-\ 1. e.: 



m- *y m. 



I. f a^ dx = mia^ = a^ '. nat. log. a, and hence: 
11. fe"" dx = e\ 

Further, from 8 (log.^x) =^ , there follows 1 '-=zlog.^x^ 

i. e.: 

rdx 1 
— = — log.^ X = nat. log. x, and the formula o (nat. log. 

ux 
x) = — gives the same. 

From this, the following examples may be easily performed: 

ye^^- ^dx = ife^''"^ 8 (5 ^ — 1) = i e^^"^ 

i ^ = I 1 — r h^ = I nat. log. (1 x A- 2). 

=fxdx -\-fdx + 2j"-^^^5^ = ^-^ X -\-2nat.log. {x — 1). 

x^'dx=^ --, leaves the 

m -j- 1' 

last integral undetermined; for if we put m = — 1, there follows: 

/ux i x^ 

— = 1 ^-1 8^ = ~- -j- a constant =: oo -|- constant; but if we 

put X =^ 1 -{- u^ and 8^ = 8w, we obtain: 

— = - — ; — = (1 — u -\- u'^ — u^ -\- u'^ . .) 8m, and hence, 

X 1 -\- u 

C'dx ^ r 8^^ /'(l_^, + ^2_^3_|_,,4 )g,, 

^07^1 + ^ 

11^ u^ u^ 

= „___+.____. + ...; 

11^ U^ 11^ 

we may, therefore, also put nat. log. (1 -{- u) = ic — — ■ -j 

^ O 4: 

H , or: 

lY. JSfat. log. x = (x — 1) — ■'^ — + -^^ — -^^ ^ -L... 

J \ J 2^3 4 ' 

By the aid of this series, we may also calculate the logarithms of 
those numbers which deviate merely a trifle from 1 ; but if it be re- 
quired to find the logarithms of greater numbers, we proceed as follows. 

If u be taken negative, the series before the last giA^es 

nat. log. (1 — ii) = — u — : 

^ ^ 2 3 4 ' 

3 



34 ELEMENTS OF ANALYSIS. [Art. 23. 

and, by subtraction of one series from the other, there results 

2i^ [_ _^ ), i. e. 

nat. log. i^-^-^J -= 2 |^ w + y + — H J ; or, if we 

make 



l^u x — 1 

= X. thereiore u =^ -, 

1 — tt ' ^ -f 1' 



Y. Nat log. x = 2 l""-^- -f i f-^-^Y + i ('^i::^Y+ . . .1. 

^ Ix-l-l^-'yx^lJ^^^x-irlJ J 

This series may also be emploj^ed for determining the logarithms 



of such numbers as deviate considerably from 1, since is al- 

X ~~Y~ J- 

waj^s less than 1. 

Therefore, also, log. (x^y) — log. x = Iog. \J^~^ )=log.ll -f- — J 

= 2 r ^ -4- 1 f ^ y 4- 1 r ^ Y-^ i 



and hence 



VI. Xo,. (. + ,) = Z.,. . + 2 [-/— + i (^ Y+ . • .] 



-2x -\- y. 



This formula may also be used to find, from one logarithm, the 

next greater. For example, 

r2 1 ^2 lA* -1 

nat. log. 2 = 2 [-—- + i . [---^} + • • -J 

^^^ -^ (3 ~r 3 • 27 ~r 5 * 243 "T" ' • V 
r 0,33333^ 

0,0000t 
more correctly, = 0,69314718. 

Mat. log. S = 72af. log. 2^ = 3 nat. log. 2 is, accordingl3^,^= 2,0794415, 
and, finally, from the last formula: 
nat. log. 10 = nat. log. (8 -f 2) 



16+2 ^ ^ V16 + 27 ^ J 



= nat. log. 8 -{- 2 

= 2,0794415 + 0,2231436 = 2,302585. 
We can also put 

nat. log. 2 = nat. log. 1 + 2 [^^-^ + i ( j-i-^J+ . . •] 

= 2 ( i + 4 • -|- + i • ]iH- • • •) = 0,693UV, further, 



AuT. 24.1 



ELEMENTS OF ANALYSIS. 



not. log. 5=naf. log. (4 -f 1) = 2 naf. Jog. 2 -f 2 \Jj4--l . ^H J, 

and lastly, nat. log. 10 = nat. Jog. 2 -L no.t. log. 5. 

(Comp. Art. 19.) 

Art. 24. The trigonometric and circular functions^ whose differ- 
entials are likewise to be found in the following, are also of prac- 
tical importance. 

The function of sine y = sin. x gives, for 5? = 0, ^ = 0, 

TT _ 3,1416 



for X 



= 0,t854 ...,y = V^ = 0,tO i 1, 



for X = y, y = 1, for x = 7:,y = 0, 

ioY x^%-^y = — 1, for a; = 2r, ^ = 0, &c.; 
hence, if we take x as abscissa J. 0, and y as corresponding ordi- 
nate OF^ we obtain the meandering curve (APBtz C'2-), Fig. 33, 
which may be extended indefinitely on both sides of A. 



Fis;. 33. 



T K 




-Y EL H N 

The function of cosine y = cos. x gives, for a? = 0, y = 1, for x = 

-J^ y = Vh foi^ X = ^, y = 0, foT X = -^ y ^ — l^ foT X = ^ tt, 
?/ =: 0, for ^ = 2 -, ?/ = 1, &c.; therefore, precisel}" the same mean- 



derina: line 



(+ 



'^i^i 



- j which corresponds to the function 



of sine, corresponds also to the function of cosine; but it is, on the 
axis of abscissas, by ^ - = 1,5T08 . . further before or behind the 
curve of the sine. 

The curves, however, which correspond to the tangentio.l and co- 
tangential functions y =^ tang, x and y = cotang. x^ are of entirely 



PjC, ELEMENTS OF ANALYSIS. [Art. 25. 

different form. If, in y = tang, x, we put a? ^ 0, |- -, i -, we obtain 
?/ ^ 0, 1, oo; and lience, a curve (A QE) which approaches nearer 
and nearer to a line running parallel with the axis of ordinates A Y 

and passing through the point I -^ ) of the axis of abscissas A X, 

but which never reaches it. If, further, we take x = — , tt, | -, we 
obtain y = — oo, 0, + oo, and hence a curve {F rz G) which ap- 
proaches the parallel lines running through i — | and (| -), ad in- 
finitum., or which has, in other words, these parallels as asymptotes. 
(Yid. Art. 11.) 

In assuming further values of x., the same values of y are repeated, 
whence also, the function y ^= tang, x will be corresponded to by 
curves (as Fr. G) which are distant from one another by tt = 3,1416 
in the direction of the axis of abscissas. 

The function y = cotang. x, on the other hand, gives for a? = 0, 

— ,— -, -, ?/ = 00, 1, 0, — 00 ; hence, to this corresponds a curve 

I K Q---L I which differs from the tangential curve only in position. 
It is also easy to infer that innumerable other branches of curves, 
as, for example, i M ^- Y, j &c., belong to this function. 

Whilst both the curve for sine and that for cosine form a continuous 
whole, the tangential and cotangential curves consist of separate 
branches, inasmuch as their ordinates for certain values of x^ pass 
from positive into negative infinity, whereby the curve loses its 
continuity. 

Art. 25. The differentials of the trigonometric lines or functions 
are given in Fig. 34, in which we have 

GA=GF=GQ = l, arc A F = x, arc F Q =-- dx, further, 
F 31 = sin. X, G 31 ^ cos. x^ AS = tang, x^ lastly, 
Q = NQ — 31 F = sin. (x -^ dx) — sin. x = d sin. x, 
GF = — ( GN — G3I) = — cos. (x -f- dx) — cos. x= — d cos. x, and 
S T = A T — ^;S^=:= tang, {x -j- 8^) — tang. ^ = 8 tang. x. 

As the elementary arc F Q stands at right angles to the radius OP, 
and the angle F G A between the two lines G F and GA is equivalent 
to the angle F Q between their perpendiculars F Q and Q, the 
triangles OPilf and QF G are similar, and we have 

Q G3f . 8 sin. x cos. x 

FQ ^ GT ^' ^' ~~d^ 



i5 ^' ^- ^^ = — T— 5 ^^1^^^- 



Art. 25.] 



ELEMENTS OF ANALYSIS. 



37 



I. d (sin. x) = COS. x . dx] likewise also: 

OP PM . — d COS. X sin. x 



, hence: 



PQ 

II. d (cos. x) = — sin xdx. 

We see from this, that slight mistakes in the arc or angle have 
j,.^ the more effect upon the sine, the greater cos. 

^ is, i. e. the smaller the arc or angle; but 
that they change the cosine the more, the 
greater si?}, x is, i. e. the more the arc ap- 

S proaches to -— ; and that, lastly, the difFerent- 

ial of the cosine has a different sign from that 
of the arc, so that, as already known, an in- 
crease of X gives a decrease of cos. x., and 
inversel};', a diminution of ^, giA^es an increase 

of COS. X. 

If SP be drawn at rio-ht angles to C I'l 
'^A there results a triangle SP1\ which, on ac- 
count of the equality of the angles PTS and CQNoi (7 P if, is 
similar to the triangle C P 31^ and hence, we have: 
ST__ CP . 8 ta7ig. X __ 1 

Yr ~ Inr ^' ^' ^ ~" COS. X 




U M 



But we have also : 
III. d(tang. x) = 



secayit. x 
dx 



1 c) X 
, hence SP = and 

COS. X COS. X 



(cos. xy- 

TV f 7Z 

If, for ^, we substitute — — ,37, therefore, for 8-^, 8 I -^r- 
— 8^, we obtain: 

8 tang. { ~ 



X \ 



X 



ex 



[co,. (-^ - ^)]' 



1. e. 



lY. 8 (cotang x) = — 



8^ 



(sin. xY 
By inversion, these formulae give, for the differential of the arc: 



8^ = 



8^ 



8 sin. X 
cos. X 



8 COS. X 



sm. X 



= (cos. xy 8 tang, x 



8 sin. X 



= — (sin. xy 8 cotang. x, or; 
8 tang, x 



V \ — (sin. xy 1 + (^^^^9- ^y 



38 ELEMENTS OF ANALYSIS. [Art. 2G. 

, ^ 8 COS. X d cotanq, x 

as also, ox = — = — ~ -. 

1/ 1 _ (^cos. xf 1 -h {cotang xf 

If now sin, x be designated by y, and x by arc. (sin. = ^/), 
we obtain: 

Y. 8 arc. (sin. = y) = ^^ ; 

and in the same manner we find: 

8 V 
YI. 8 arc. (cos. = y) z= — - — ^ , as also: 

y 1 — ?/^ 

YIL 8 arc. (tana. = i/) rz= — , and 

V y J^ 1 + ^' 

8 ?/ 
YIII. 8 arc. (cotang. z= y) = 



1 + y^ 

Art. 26. The last differential Ibrmnlae give, by inversion, the 
following integral formnlae: 

I. J COS. X d X = sin. x^ 

II. f sin. X d X = — COS. x, 

dx 



III. I = tang. x. 

' COS. x^ 



lY. j— ^ — ' — ■ = — cotang x. further: 
J sin. x^ 

Y. f — ^^= = arc. (sin. = x) = ■ — arc. (cos. = x). 

JVl — x' ^ 

C c)x 
YI. |- — - — ^ = arc. (tang. = ^) = — arc. (cotang. = x)^ 
«y i — r- X 

and the following may also be easily found. 

^^^ -, r^ , 7 . N 8 sin. X COS. X . d X 

We have c (nat. log. sin. x) = ■ — -. = -. • = cotang. 

* sm. X sin. X 

X . dx, consequently: 

YIL fcotg. xdx ^= nat. log. sin. x, likewise: 

YIII. ftang. xdx = — nat. log. cos. x, further: 

dtang.x dx 

c (nat. log. tang, x) = = 

tang, x cos. xr tang x 

dx d (2 x) ' 

^= —r- = —. — ^-, hence: 

sm. X COS. X sin. 2x 

8 X 
8 (7iat. log. tang. ^ x) = ■—. , and 

o t/t. X 

C dx X 

IX. \—. — — = nat. log. tang. -— , likewise: 
«y Sin . X jj 

^^ f dx , ^ , f ^ , ^ ~\ 

A.. I = 7iat. log. tana. ( h "t^ I 

J COS. X -^ ^V.4"27 

= nat. log. cofg. 1-— ■ — ^ )• 



[Art. 27. ELEMENTS OF A:N^ALYSIS. 39 

If, further, we put -„ = -z— \- = -^ ^— — -^ — - — -, 

there follows I = a {I — x) -f 6 (1 -f ^). If we take 1 -f ^ = 0, 
therefore, x = — 1, we obtain 1 = a (1 -f 1), as also « = -1; and 
if we put 1 — X = 0, therefore, ^ = 1, there results 1 = 26, hence; 

111 

6=1 and = -— ^ — ■ -f —^ — , 

^ 1 — x' l-^x ' 1 — x^ 

but lastl}^: 

r dx _ J r dx ^ r dx 

J i—x^ — 2 J Y^x ^ V l — x 

= J nat. log. {1 -{- x) — ^ nat. log. (1 — x)^ 
i. e.: 

/ox /^ 1 I ^~\ « 
= 1 nat. log. 1 |, and likewise: 
1 — x^ VI — xj 

-t^-r-r r 8'^ n 7 fx — 1~\ 

-^"- j ;^i=T = i "«'• '^3- l^Ti J- 

If we put l/l -\- x"^ = xij, we obtain 1 -\- x^ = x"^ 2/^ ^^^^ 
S ^ (1 — y-) = xyd y, hence: 

— ^ = - — —, = i 8 nat. log. \ , }, from which 

there follows 

XIII. ( . = nat. log. (x + j/i i ^i--), as also, 

^ V 1 -{- x'^ 

/ ., = ^^«^^. ^OS'- (^ + Vx' — l)- 

^ — -^ it is only neces- 
sary to develop into a series, by means of division, and then 

-^ ^ I ^ x' ' -^ ' 

integrate each member. Thus we obtain 

=1 — x''^ -\- x^ — x'^ 4- x^ • •, and 

1 -j- ^ 

I ^g =: { cx — I x'^-dx -[- I x^dx — j x^'dx + • • •, 

consequently : 

mO /y»5 rp* 

I. arc. {tang. = x) = x — -\ — ^ -j- • • •« for example: 

-J- = (^rc. {tang. = l) = l_i-fi_j + | , • 

therefore, the semicircle - = 4(1 — ^ -\- I — |-j- i ..), or, 

^ = arc. {tang. =-,/|) = y^ [1 _ i . ^ -f i {Vy _ , ^ly _^ . . .], 
consequently, n = 6 y^ (1 _ i + ^V _ ^i^ + . . .) = 3,1415926 • • • 



40 ELEMENTS OF ANALYSIS. [Art. 28. 

In like manner we obtain 

^ = (1 — ^0""' = ^ + i ^' + I ^- + i^F ^' H 






i. e.: 

,1^=^ , 1.3x'' , 1.3.5if?^ , 
IL ^-^.(^•^^• = ^)=^ + 0^- 27475 + 2X6:Y+-"' 
for example: 

- = arc. (sin. = i) = i (1 -[- 2¥ + ^lo + ttVs + * ' O? 
therefore : fl, 041671 



TT 



0,00469 1 
^ ^-^0,00070 ^=^'^^^^- • 



0,00012 
There follows, further, from successive differentiation, if we put 

si?i. X =^ A^ -\- A^ X -{- A^ x^ -j- A.^ x^ -\- A^ x^ -j~ • ' ' : 
d (sin. x) 



= COS. X =^ Aj^ -^ 2 A^ X -{- S A^ x'^ -\- 4. A^ x^ -{- ' 
= — sin. X = 2 A,^-i- 2 .S A^x ^ ^ . 4: A^x^ ^ 

= — COS. ^ = 2. 3. ^3 + 2. 3. 4. ^^^4-.. 

= sin. J7 = 2.3.4.J+... 



dx 
d (cos. x) 
dx 
d (sin. x) 

dx 
d (cos. x) 

dx 

But we have for ^ = 0: sin. ^ = 0, and cos. x = 1] hence, there 
follows from the first series: ^^ = 0; from the second: A^ = cos. 

= 1; from the third: ^^ == 0; from the fourth: A„ = — - — ; from 

2.3 

the fifth: A^=^ 0, &c.; and if these values are substituted in the as- 
sumed series, there results the series of sine: 

HI. sin. a^ = j- j^73 + 1.2.3.4.5 - I .2.3.4.5.6.T + ^'^ 

In like manner w^e have 
IV. COS. ^ = 1 - i^ + ^-^ - -^^-^^ + &c, further, 



x^ , 2 x'" , 1^ X 



^^_ 1 X x^ 2 x^ 

VI. cotang. ,, = ^ _ - _ _ - _ ^-^-^ _ &o. 

Art. 28, If the differential formula d (uv) = udv -j- vdio of 
Art. 8, be integrated, there results the expression uv = fudv -|- 
fvdu^ and the following integral, known under the name, reduc- 
tions^ formula : 



[Art. 28. ELEMENTS OF ANALYSIS. 41 

fv'du = uv — fudv, or, 

f<p (x) df(.x) = <p (x)f(x) —ff(x) d <p (x). 
This rule is always applicable when the miegvalfvdu ^f 9 (^c) 
of (x) is not, but the integral /* z( S u :^ ff(^) 8 9 (^) is, known. 

lyy means of the reductions' formula, the integral from the follow- 
ing differential: 

dy =z y I -]- x'^ . dx^ 
may, for example, be reduced to another known integral. "We have 
to put 

, (f, 7) X 

(p (x) = y 1 -|- x'\ therefore, 8 (p {x) = — - -— ' , and 

1/14- X'' 
f (x) = X, therefore, df (x) = dx: 

consequently, ^Ye have 

r / r x''dx 

I V 1 A- x' dx = X V 1 -'r ^' — I "7"7^=,' 
but: 

x' _ _ l-\- x^ 1 / , _ 1 

j/'T^^x" ~ V i-^x' 1/1"+"^^ ~ ^ 1 -I- ^'' y'-i:^i' 

hence, there follows 

J i/T+^^ 'dx = x i/ir^^ —J /T -f- x'' dx -[- J -/--'y==,'> 

^ J. ~Y' X 

or: 

2 I V 1 + a^ dx = x V"l -f x' + I -y-^^=, 
and consequently; 
I- J V~l^ x'dx = ^x V l + x' + ij 77Y 

= ^ [^ l/"l 4- ^2 + nl. {x 4- T/^lTf^)]; 
likewise : 

II. J ]/ 1 _ ^2 g ^ ^ 1 ^ l/l_^2 _|_ ^ J .^^^^ 

= I- l_x V 1 — ^^ -f a^^c. (6'Z«. =^ ^)], 
and 

III. I l/ 0^2 T Pl-y- 1 ^ l/^2 T 1 



8j7 



-II- J l/ ^^ _ 1 8 j; = 1 ^ l/ ^2 _ 1 _ 



2 J ■/^•^_1 

= i [.^ l/ ^'2 _1 — nl. (x -\- V x' — 1)]. 
We have also 

J (sin. x)'^ dx=J^sin.xsin. xdx = — fsin. xd (cos. x) = — sin. xcos.x 
-\-fcos. xd (sin.x)= — sin. x cos. x -\-/(cos. xydx 

= — sin.xcos.x^f[l. — (sin.xy] dx, 
hence, there follows 

'2 J (sin. xydx =fd x — sin. x cos. x, and 



42 



ELEMENTS OF ANALYSIS. 



Art. 29.1 



IT. f (sin. xydx = ^ (x — sin. x cos. x) ^= ^ (x — i sin. 2 x). 

There is, likewise, 
Y. f (cos. xy dx = ^ (x -^ sin. x cos. x) = ^ (x -\- ^ sin. 2 x). 

Y"e have, further, 
YI. f sin. X COS. xdx =-- ^f sin. 2^8(2^)= — \ cos. 2 x^ 

YII. f (tang, xy dx ^ tang, x — ^, and . 

YIII. f (cotg. xydx =z — (cotg. x -f x). 

Finally, there is, 

IX. f X sin. xdx ^^ — x cos. x-\~f cos. xdx = — x cos. x -{- sin. x, 

X. fxe^'dx =fxd (e^ = xe^ —f^'' dx = (x — 1) e% 

dx 



XL I nat.Iog. X cdx = xnat.log. X — | x — - = x (iiat.log. x — 1), 

x^ C 

(x nat. log. xdx) =-^ nat. log. x — | 



and 
XIL 



x^^ dx 



= (nat. log. x 



i) 



x"" 
~2 



Fia'. 35. 



D 





Art. 29. If it be required to quadrate a curve AP B., Fig. 35, 
i. e., to determine or express the area of the surface ABC, which is 

bounded b}^ the curve AP B and its co- 
ordinates A C and i? (7, by a function of 
the abscissa of this curve, let us imagine 
this surface to be distributed by an in- 
finite number of ordinates J/P, YQ, &c., 
into laminated elements, as IIJSfQP, of 
the constant breadth MN= dx and the 
variable length 31 P = y. Since we may 



^,^a^ 



s.x%-^s^<»s^^^ 



Mx\ 



C 



now put the area of such an element of surface, 

^ -J. J/Y=(y-f I dy) dx = ydx, 



g^=(5^ + ^^«^ 



the area of the entire surface F may be found by integrating the 
differential ydx, thus putting 

F =f ydx. 
For example, for a parabola with the parameter p, we have y'^ =px, 
and hence the surface of the same: 

/ 3 

V~p^dx = Vpj x''dx = —-3--- = I ^ V px^ 



F 



xy. 



The parabolic surface AB G is., therefore, two-thirds of the rec- 
tano-le AG B 1) which encloses it. 



Art. 29.] 



ELEMENTS OF ANALYSIS. 



43 



This formula is also applicable to oblique angled co-ordinates 
intersecting each other at an angle XA Y = a] for instance, to the 
surface AB G^ Fig. 36, if, instead of B G = y, the normal distance 
B N = y sin. a be substituted ; we have here, therefore, 

F ^= sin. a f ydx. 

For the parabolic surface, for example, when the axis of abscissas 

A X forms a diameter, and the axis of ordinates AY^Si tangent, of 

px 
the parabola, and therefore, y"^ = p^ x = 



sin a 



2 5 



there results 



JF =z ^ xy sin. «, 



1. e.: 



surface AB G 




I parallelogram AG BD. 

Fig. 37. 




For a surface B G G^B^ = F^ between the abscissas AG^ = C-^^ and 
A G =^ c^ Fig. 3t, there is, according to Art. It, 

For y = — , there is, for example: 



F = 



"^ a'^dx 



X 



= a^ (nat. log. c^ — not. log. c), i.'e.: 



■F = a^ nat. log. I — - i. 

The curve P Q, Fig. 38, with which we have become familiar in 

a^ 
Art. 3, corresponds to the equation —-, and hence, if we have ^if^c 

and AN = c. 



(f) 



F = a"^ nat. log. I — - 
" ^ c . 

will give the area of MX Q P. If, for the sake of simplicity, we 
assume a = c = 1, and c^ == ^, we obtain 

F — - nat. log. x^ 
and the areas (IJ/Pl), {INQl)^ &c., are the natural logarithms 
of the abscissas AM^ AN^ &c. The curve itself is an equilateral 
hyperbola., in which the two semi-axes a and b are equal, conse- 
quently the angle of asymptotes a r= 45^, and the straight lines A X 



44 



ELEMENTS OF ANALYSIS. 



[Art. 30 



and A F, to which the curve approaches nearer and nearer without 

reaching them, are the asymptotes of the same. On account of this 

Fig. 33. connection between the abscissas 

and the areas, the natural loga- 
rithms are frequently called hy- 
pe7^bolic logarithms. 

Art. 30. Every integral 

may also be put equal to the area 
of a surface i^, and if the integra- 
tion cannot be performed b}^ one 
4 ■^- of the known rules, the integral 
may be found, at least approximately, if the area of the correspond- 
ing surface be ascertained by the application of knovv^n geometrical 
rules. 

Ym. 39. 




V'- 



B 



A 



For a surface ABP QN^ Fig. 39, which is deter- 
Q mined by the base AN=x^ and the three equi-clistant 
ordinates AB ^ ?/^., ATP = ?/j, and N Q = y.^^ we have 
the trapezoidal portion 



ABQN=F, = (^j,-{-y,) 



X 



M 



-N 



I PS . BR 



§ {3IP 
hence, the entire surface: 
F 



and the segment formed portion BPQSB^ if BPQ 
be regarded as a parabola, 

2/o + y. 



US) .AN 



^iijM-^^'} 



X, 



-f': + -f;= I i (i/o + 2/.) + § f ?/, - ^— ^ 



X 



X 



= [f (Vo + 2/2) + I yj ^ = (yo -r ^ yi -}- y^) -j- 

If we introduce a mean ordinate y^ and put F = xy, we obtain 
for the same: 

!/o + 4 y, + ?/2 



y 



6 



In order now to find from this the area of a surface 31 A B TV, 
Fig. 40, which stands upon a given base MN= x^ and is determ- 
ined by an uneven number of ordinates y^^ ?/,, ?/„, y.^ .. y^^ being 
distributed by these into an even number of strips of equal l)readths, 
it is only necessary to repeat the application of the last rule. The 

X 

breadth of one strip is = — , and from this the surface of the first 

n 

couple of strips: 



Art. 30.] ELEMENTS OF ANALYSIS. 

Fig. 40. 2/0 + 4 y^ + 2/2 2.57 



45 




6 



n 



M^ 



of the second couple: 

!/2 + 4 2/3 + ?/4 2.r 



6 



n 



of the third couple : 



N 



?/, + 4 !/5 + y, 2j7 ^^ . 
6 - ' •' 



n 



therefore, the area of the first six strips, or first three couples, is, 
since we have here n = 6, 

-^ = (!/o + ^ 2/. + 2 2/, + 4 7/3 + 2 7/^ -f 4 7/^ + y,) 



3.6 



X 



= [2/0 + 2/6+4 (y^ + 7/3 + y^) + 2 (?/^, + t/J] YgJ 
hence, it is easilj^ inferred that the area of a surface distributed into 

four couples is 

so 
^= [2/0 + 2/8 + 4 (2/, + 2/3 + ?A + 2/t) + 2 0/, + 2/, + 2/6)] ^^ 

and that, generally, the area of a surface of n strips ma}^ be pnt 



a: 



F= [2/0 + 2/. + 4 (2/, + 2/3 + - + 2/._x) + 2 (2/, + 2/4 + - +.2/n-2)] 3^- 



y 



The mean height of such a surface is : 

^ 2/0 + 2/n + 4 (2/, + 2/3 H h 2/n-l) + 2 (2/, + 2/.I H V Vn-d 

3n ' 



in which n must always be an even number. 

This formula, known by the name of Simpson^s Hide, is appli- 



cable in determining 



an 



ydx = j tp {^x) 8 5?, when we distribute a? = c^ — c 



into an even number n of equal parts, calculate the ordinates 

f x^ C 2:27"\ 

2/0 = ^ W^ 2/1 = ^ ( ^ + -J, 2/2 = ^ (^^ + — J^ 



2/3 



= ^ (^ + ^1 • • ^ 2/n = ^ (^)5 



'3 r n - I ^,9 

and introduce these values into the formula: 



ydx = j o (^) 8 X 



-= [2/0 + 2/n + 4 (2/1 + 2/3 ^ h 2/n-i) + 2 (2/, + 2/, H F 2/0-2)] -kr* 



71 



J 2 g^ 
— gives, since we have here c^ — c = 2 — 1 = 1 

1 X 

and y = (f (x) = -, if we assume ?i = 6, and consequently - = 



1. 
6 • 



46 



ELEMENTS OF ANALYSIS. 



[Art. 31. 



0,5454, and y 



y,=\ = 1,0000, y^ = l=^ = 0,85n,7, = ] = i = 0,T500, 

ys = l = ^ = 0,6666, y, = ~^ = 0,6000, y, = ^- 
= 0,5000, hence: 

Vo + Ve = 1.5000, y^ + 2/3 + 2/5 = 2,0692, and y^ + y^ = 1,3500, 
and the mtegral songht: 

J ^ - = (1,5000 + 4 . 2,0692 + 2 . 1,3500) . ^V^ ^^^ ^ 

From Art. 22, III, Tve have 



0,69315. 






= nat. log. 2 — nat. log. 1 = 0,69314t 



therefore, the conformity is as desired. 

Art. 31. In the seqnel ^q shall give another rule, which may 

also be employed when the number 
n of strips is an uneven one. If a 
small segment A MB., Fig. 41, be 
treated as a parabolic segment, there 
follows, according to Art. 29, for the 
F=% AB . J/D, 
or, if AT and B T are tangents to the ends A and B. and hence 




CDE 

area of the same : 



GT= 2C3I: F=% . 



AB . TE 



3 • 9 — 3 



= I of the triauo-le A TB 



of the equally high isosceles triangle A SB; and therefore also = | 
AG , 05 = 1 AG', tang. SA G. The angle SAG = SBG is = 
TAG-}- 'TAS= TBG— TBS; hence, if we put the small angles 
TA S and TB S equal to each other, we obtain for the same : 



TAS= TBS = 



TBG — TAG 

2 



and 



SAG= TAG -{- 



TBG — TAG TAG -^ TBG d ^ s 



2 2 2 ' 

if the tangential angles TA G and TB G are designated b}^ d and e. 
Since we have, further, AG = BG =- ^AB= J chord s, we have also, 

F=is^tang.(J-±--y 

This formula may now also be applied to the portion of surface 
MABN, Fig. 42, whose tangential angles TAD = a and TBE=^ 
are given. If, for example, we put the angle of QhoYd.BAD=z 
ABE = <T^ Tie obtain 

TAB = d = TAD — BAD = a — <>, and 
TBE = e = ABE — TBE = <7 — ,3, hence: 

d -\- £ = a — iJj 

and the segment upon A B^ 



[Art. 31. 



ELEMENTS OF ANALYSIS. 



47 



XoTT, as « — /? is veiy small, 

F=^ tang, (a-^ = 'L (J^^9,- -tang. ^X 
12 ^ ^ ' ^ 12 VI _ tang, a tang. fJ' 

or, since a and y5 do not deviate largel}^ from each other, and hence 
may be replaced in tang, a tang fi hy the mean yalne (t, v 

tang, a — tang. ,3 



F 



T2 



W 8' 



tang, c^ 



= -jL- s^ COS. (j^ (tang, a — tang. ,3). 



Fig. 42. 




Therefore, hj substituting for s cos. a 
the base MN = x^ we have 



X' 



and hence the entire i^ortion of surface 
MABN., if y^ and ^/i represent its ordi- 



nates JIA and iV^5 : 

X 



^i = OJo + 2/i) f + (^«"^- ^^ — ^^^^'^ y5) 1^ 



If, with this portion of surface, there is an adjacent portion NBCO 
haAdng a like base NO ^ x., the ordmates ^ i^ and G =^y^ and y^, 
and the tangential angles SBF= ,-i and S C G =^y^ the area of the 
same will be 

■^2 = (!/i + 2/2) 2 + (^^^^^* '^ — ^^"^- "^) 12' 

and hence the entire area, since here — tang. ^5 and -j- tang. ,3 cancel 
each other: 

F= F^^ F^=(^ !/o -rVi-T- i Vd ^ ^ Qung. a — tang, y) — . 

We have, likewise, for a surface consisting of three ecjuallj^ broad 
strips, if a represent the tangential angle of the initial, and that 
of the terminal, point: 

^ == (i- 2/0 + 2/1 + 2/2 -r i- yd ^ + (^«^'^5'- « — iang. d) — , 

and generall}^, for a i^ortion of surface determined by the abscissas 

X 2x 2>x 

-, — , — • • • 5?, the ordinates ^^, ?/^, y,^ • • • !/^5 and the tangential 

anoles « and a of the terminal points: 

■—On -I- 



^=ay, 



r !/i + 2/2 -r 



X 



2/n-l + i 2/n) - 



?1 



+ tV Q(^^^q- « — ^^"5'- O OD * 



48 ELEMENTS OF ANALYSIS. [Art. 32. 

An integraL- 

c 1 /'ci 

ydx =j (p (x) da; 

=- (i 2/o -f !/i + 2/2 + • • • -h2/„-i + J yj ^ 

is found, accordingly, if we put x =^ c^ — c, and calculate 

as also tang, a ■= ^ =^ 4, (^t) ^ <!' (c) and tang, a^ = 4> (c,), and 
introduce these values into this equation. 

Thus if we assume ?7 = 6, we obtain, for I --^, since there is here 



tang. a = — \ = — 1 and tang. (3 =z — | - j r= __ J, 



•^1 X ^ 

x = c^ — c = 2—l and y = o (a:) = ~: 

X 

^0 ^' -"-J !/i — -j^ I j^ — Y7 !/2 — 8 1 2/3 — SS 
2/4 = i%i 2/5 = IT, ^^nd ?/,^ = T% ; 

further, as there results ^ = £_xL_Z -_ . 

ex ox x'^ 

■2. 
and hence there follows 

J- f)jfj 

4.1692 

= -—, I • T2 • B6 = 0,69487 — O.OOltS = 0,60314. 

b . ' 

(Comp. the example of the foregoing article.) 

Art. 32. Li order to rectify a curve, or, from its equation y ^= f 
{x)^ between the co-ordinates AM=- x and MP = y., Fig. 43, to 
deduce an equation between the arc AP = s and the one or the 
other of the two co-ordinates, we must first determine the differential 
of the arc A P, and then seek tlie integral. If x increase 'bj MN = 
P Pi = dx^ y will increase by i? ^ = 8?/, and s, by the element P Q 
::= 8s; and we have, in accordance with the Pythagorean theorem, 

P'Q' = P~P' + Q R\ i. e.: 8s^ = dx' + 8?/'; 
therefore, 

8 6" = y'dx"- -\' dy'\ and consequentlj^, the arc itself; 



s 



Vdx' 4- dy\ 



Art. 32.] 



ELEMENTS OF ANALYSIS. 



49 



For NeiVs parabola^ for example, (yicI. Art. 9, Fig. 1*1,) the equa- 
tion of which is ay''- = x^, we have 2aydy = Sx"dx; hence: 



according to which, 



Sx^dx , ^ , 9 x' dx' 
and. cy^ = 



2 ay 4: a'^ y'^ 



4:a 



(^ + ia) ^^'' """"^ 



-/^''+^>-?/o+^D'»(^:) 



^ -1 



In order to find the constant which is here necessary, we will 
allow s to begin simultaneously with x and y. AYe thus obtain 
= 2T a y 1 ^ -f con., therefore con. =^ — /y ^j ^^^l 

for example, for the portion AP^, having the abscissa x ^ a: 
^ = 2\ci [i/(W — 1] = 1,^36 a. 

Fi''. 43. 

If, further, the tangential angle 
QPR = P TM= a be introduced, 
there results also, Q R ^ P Q . sin. 
QPR v.nd PR = PQ COS. QPR, 
i. e.: 

dy = ds sin. a and dx = 8s cos. a, 
and therefore, not only 

tang. a = ^ (Art. 6), but also 




M N 
dy 



dx 



sin. a = —- and cos. a = _- ; and, further, 

OS cs 

dy 



^ = j t/ 1 + tang. -^ -d x =j^^^^ =j ^^^. 

If, now, the equation between two of the magnitudes x, y, s, and a 
be given, we can then also find equations between two others of 

these magnitudes. If we have, for instance, cos. a =-. ,, 

' ' Vc''-\-s'' 

there is also; 

ex = OS COS. a = 



du 



and 



J l/c' -4- s' -^ Vc' 4- s' -^ Vu ^ 



Vc' + 



50 ELEMENTS OF ANALYSIS. [Art. 33. 

= ^/c^ -f s^ -f- con.; and if x and s are, at the same time, zero: 

X == Vc^ H- s' — c. 

Art. 33. A straight line at right angles to the tangent P T, 
Eig. 44, is also normal to the jDoint of contact F of the curve -, be- 

Fig. 44. 




T A K 2f ^tC -^^ 

cause the tangent indicates the direction of this point. The portion 
P K of this line between the ]3oint of contact P and the axis of 
abscissas, is called simply, the normal^ and the projection 31 K of 
the same on the axis of abscissas, the subnormal. We have for 
the latter, since the angle 31 FK is equal to the tangential angle 
F T3I=a, 

. 31 K = 31 F . tayig. a, i. e.: 

dij 



the subnormal = y tang, a ■= y 



d^r' 



As, for the system of curves y = x"^, there is tang, a 
follows here the subnormal = mx^ . x^"'^ 



mx'^ ^, there 



m.x 



2"^~i = -— ; and for 

X 



the common parabola., whose equation is y'^ ■= px, we have 



the subnormal = y 



p_ 
22/ 



P. 
2' 



therefore constant. 



If, further, another normal ^ C be drawn at a second point Q., 
infinitely near the point P, there results, in the point of intersection 
of the two lines, the centre of a circle to be described through the 
two points of contact P and (5, or the circle of gyration] and the 
portions OF and G Q of the normals are the radii of this circle, or 
the radii of gyration. Of all the circles which may be drawn 
through P and Q, this is the one that most nearly coincides with 
the circular element P Q^ and we may therefore assume that its arc 
P Q coincides with that element. 

If we designate the radius of gyration CP = C Qhy r, the cir- 
cular arc AF by s, therefore its element P (^ by 8 s, and the tan- 
gential angle or arc of P T31 by a, therefore its element S U31 — 



Art. 33.] 



ELEMENTS OF ANALYSIS. 



51 



S TM, Le.— UST= — PCQ, by 8«, we have simply, since there 

is F Q = G P . arc of the angle P C Q: 8s = — r8aj and couse- 

8s 
quently , the radius of gyration : r = — ^r-. 

Fig. 45. 




^^ T A M if K 

Generally, « can only be determined by means of the equation of 



co-ordinates, since we put tang, a = -^, 

ox 

But there is, further, '8 tang, a = 



COS. a? 



dx 
and COS. a ■=--z^\ 

8s ■ 



8 x^ 
hence, we have 8« = cos. a? . 8 tang, a == — — . 8 tang. «, and 

G o 



8s 



8s^ 



cos.a^dtang.a dx'^dtang.a' 

ds ds^ 

For a convex curve, we have r = -\- ^z-- = -{- - — 

0(j^ cx"^ c tang, a 

for a point of inflection, r 



; and 



oo. 



For the co-ordinates AO = u and G = v of the centre G of 
gyration there is 

u = AM -\- HG = X ^ G P sin. G P H^i. e. u = x -{- r sin. «, and 
v= 0G = 31P — HP = y—GP cos. G P H, i.e. v = y—rcos.a. 

The continuous succession of 
the centres of gyration gives a 
curve which is called the evolute 
of A P, and whose course is de- 
termined by the co-ordinates u 
and V. 

If the ellipse ADA^D^ Fig. 
46, be brought into connection 
with a circle AB A^B^^ its co-or- 
dinates GM= X and MQ = y 
may be expressed by the angle 
P G B = ^ at the centre of the 
circle. We have, for example, 




52 ELEMENTS OF ANALYSIS. [Art. 33. 

X = CP sin, OF 31= GP sin. BCP = a sin. <p and 

V = MQ = - MP =- GP COS. CP3I=b cos. cp. 
^ a a 

From this tliere results 

^x = a cos, (f'b(p and dy -= — h sin. (p'd(p^ 

consequently, for the tangential angle Q TX = a of the ellipse: 

dy h sin. cp h 

tana, a = — - = — = tang. <f. 

ex 'a COS. (f a 

and, for its adjacent angle Q T G =^ o.^ r= \^{)^ — a: 

b ^ a 

tang. a^=z - tang. and cotg. 0.^ = — cotg. (p. 

Accordingly, the suhtangent of the ellipse is 
MT= MQcotg. MTQ 

= y cotg. a^ = --- cotg. <p = y^ co^- 9, ' 

if y^ designate the ordinate IIP of the circle. Since, in the latter, 
the tangent P T stands at right angles to the radius CP, there is 
also P TM = P G B = <f^ and hence, the sub-tangent of the same, 
likewise: MT= MP cotg. 31 TP = y^ cotg. cp. Therefore, the two 
points P and Q of the circle and of the ellipse which have the same 
abscissas, have one and the same subtangent 31 T. 
We have, further, for the elliptical elementary arc: 

8s^ = 8^^ -|- 82/^ = {oL^ COS. (p'^ -\- b'^ sin. <p^) 8c^^, 
and the differential of tang, a : 

c tang, a = c tang. ^ = ; 

a a COS. cp"^ 

hence, there follows the radius of curvature of the ellipse: 

ds^ (a^ COS. (p'^ -f- 6^ sin. w'^y^ 

dx^ d tang, a „ „ 6 

a^ COS. ip^ . - 

a COS. (p^ 

(a^ COS. (p^ -\- b^ sin. ^^) -^ 

ab 

For example, for ^ = 0, therefore sin. <p=^0 and cos. <p = l, there 

follows the greatest radius of curvature: 

a^ a"^ 

'^'^~ab~ y 
and on the other hand, for <p = 90'', therefore sin. ^ = 1 and cos. 

^ = 0, the smallest radius of curvature: 

_b' _b'' 

" ab a' 
The first value of r corresponds to the point D, and the last, to 
the point A; both are determined by the portions of axes GL and 
(7/f, which, from C, cut off at the extremities A^ and D the perpen- 
diculars erected upon the chord A^^ D. 



[Art. 34. ELEMENTS OF AT^ALTSIS. 53 

Art. 34. Many functions which occur in practice may be com- 
posed of the principal functions treated of above : 

y = ^™, y = e^\ and y = sin. x^ y = cos. x, &c , 
and it is also easy, by aid of the foregoing, to find their properties 
with respect to the position of tangents, quadrature, radii of cur- 
vature, &c., as also, to construct the curves corresponding to them, 
as will be shown in the following example. 

For the curve corresponding to the equation: 

y =z x^ i 1 — - J = a;^ — J ^^, we have 

dy = 2xdx — x'^doc^ consequently, 
tang, a = 2x — x^ = x ^2 — x). 
As this tangent is = for x = and x = 2^ it has, also, at the 
points which appertain to these values of abscissas, the direction 
of the axis of abscissas. There is further: 

d tang, a = 2dx — 2xdx = 2 (1 — x) dx, 
according to which, we have 

for X = 0^ d tang, a = -{- 2dx^ and 
for X = 2. d tang, a = — dx; 
and hence, the ordinate of the first point is a miiiimum^ whilst that 
of the second is a maximum. If we put 8 tang, a = 0, we obtain 
thereby, the co-ordinates, x = l and y = ^-, of the point of inflection 
at which the concave portion of the curve joins the convex portion. 

We have, further, for the elementary curve 8s: 

8s2 = dx'' -]- 8?/' = dx"" -^x'' {2 — xf dx^ = ll^x'' {2 — xy] dx\ 
and hence the radius of curvature of the curve : 

_ 8g' _ _ [i^x'' {2 — xy]' 

^ ~~~ dx''dtang.a~ ~ 2(1 — ^) ' 

for examj)le, 



— 1 2'- 

for X = 0, r = — —— = — 1, for x = 1, r = — 

2i '' ^ 



CO, 



for x = 2,r=^ = -f i, for a; = 3, r = i . 10' = + 1,906. 



, 1 . . „ „ , ^A 

, —2 

The corresponding curve is shown in Fig. 41, in which A represents 
the origin of the co-ordinates, and XX, FF, the axes of the co-ordi- 
nates. To the first part, y^ = x\ of the equation, corresponds the 
parabola BAB^^ which, from A^ passes sj^mmetrically to both sides 
of the axis A Z; and to the second part, 3/2 ^ — i ^^ corresponds 
the curve GAC^^ which, on the right side of FF passes below the 
axis of abscissas XX. whilst on the left, it passes above XX^ and 
withdraws farther and farther from the same the more it is removed 
from YY, To determine, for a given axis of abscissas x, the cor- 



54 



ELEMENTS OF ANALYSIS. 



Art. 34."] 



Fie. 47. 



responding point of the curve y = x^ — ^x\ it is on\j necessary to 
add algebraically the ordinates of the first curves, which appertain 

to this abscissa. Af we have, for 
example, for x=l^ !/i = l and 
y<i= — ^, there follows the cor- 
responding ordinate of the point 

^^ !/ = ?/, +2/. = 1—^-1; 
further, since we have for a; ^ 2, 
y^ = i and y^= — f , there follows 
also the co-ordinate of the point 
31: y = 4: — f = |. Likewise, 
we obtain for x = 3, y = y^ -j- 
y^ = d — 9 = 0,forx = i, y = 
16 — 6_4 ^ _ 1^% for ^ = _ 1, 

2/ = 1 + ^ = I, for ^ = _ 2, 
2/ = 4 -f- I = 2_o^ ^(3., and we 

perceive that the last curve has, 
from A to the right side, the course 
AWIIKL,..^ at the commence- 
ment of which it passes along 
above the abscissa AK=S^ but 
that after the point K it runs be- 
low XX indefinitely, whilst to 
the left of A, it ascends constant- 
ly, forming the indefinite branch 
AF Q .., From the above also, 
"FT is a point of inflection, and M 
a maximum point, of the curve. 
Whilst at A and Jf the curve has 
the direction of JTX, at W it 
ascends at an angle a = 45*^, be- 
cause we have for the same ^tang. 
a = X (2 — x) = 1; but for the 
angle of inclination at K^ there 
is tang, a = — 3, consequently, 
a = 11' 34^ &c. 

The quadrature of the curve 




_X 



X' 



X 



is performed by the integral 
F= fydx = C(x^ — 

x^ X* ^^ r 1 ^ 

"" T ~~ 12 "" TV "~ 4. 
Accordingly, there follows, for example, for the portion of surface 



') dx= Cx'dx^ljx'd 

)• 



[Art. 35. ELEMENTS OF ANALYSIS. 55 

33 
A WMK above AK=^, the area i^= — (1 _ |) = |, and, on the 

other hand, for the portion 3X4 below the portion 34 of the abscissa, 
i^, = y(l-|)-|-(l-|) = 0_f = -|. 
Lastly, to find the length of a portion of a curye, as A W2I^ we put 

s =J V 1 -^ x' (2 — xf dx = P' <p (x) dx, 

and apply the method of integration discussed in Art. 30. We have 
here c ^ and c^ = 2] if we assume n = 4, there follows dx = 

^^^^ = ?^^ ^ 1 , and if in the function c (x) = Vl-{-x' (2 — xY, 
n 4 

we substitute for x the values 0, i, 1, f , 2, successively, there will 

result: 

^ (0) =: 1/T = 1, ^ Q) = i/iT:S = h 

cp (1) = v^l-l-l =.: 1/2 = 1,414 . . . 

^ (I) = lT+TV = I, and c. (2) =.1/1 = 1, 
and hence the length of the arc A WM: 

s = [<P (0) + 4 c. (1) + 2 c. (1) + 4 c. (I) + ^ (2) j ^^ 
= (1 -f- 5 + 2,828 + 5 + 1) . i = 2,4n. 

By means of the curve y = x^ I 1 — - j , the course of the curve 

/ X 

y = 00 -xl 1 — - may now also be easily indicated; for if we extract 

the square root of the co-ordinate -values of the first, there result 
the corresponding co-ordinates of the last. As the square roots of 
negative quantities are imaginar}", this curve does not extend be- 
yond K; and since every square root of positive quantities has two 
equally great and opposite values, the new curve (II.) consists of 
two symmetrical branches Q A MX and Q^ A M^ K, one on each 
side of the axis XX, 

Art. 35. When the quotient y = ) ) ( of two functions c> (x) 

and ^'' (x) assumes, for a certain value a of x^ the indeterminate 

^2 ^2 

value -, — which is always the case when, as in ?/=- — , the nu- 

\J tXj ^'~~ (Jj 

merator and denominator of a fraction have one factor x — a in 
common — , the real value of the same may be arrived at by diifer- 
entiating the numerator and denominator separate^. 



56 ELEMENTS OF ANALYSIS. [Art. 35. 

If ^ increase by the element doc^ and y^ correspondingly, by the 
element 8?/, there results: 

But we have now for x = a: 

(p (^x) = and (p {x) = 0; 
hence, there is for this case : 

or, since 8?/ as infinitely small magnitude vanishes in comparison 
with y: 

^ (p(x) ^ dcpjx) ^ cp^ {x) 

'^ ~ (/'(x)~ d 4' {x) 4\ {^y 

in which (p^ {x) and 4'^ {x) designate the differential quotients of 

<P (x) and 4^ (x). 

<P (x^ 

If y = ^ y is again = -, we can differentiate anew, and put 



The undetermined expressions y =-—', . oo, &c., may be treated 

1 
0' 



GO' 
1 

in like manner, since we may put oo = -, consequently -^, and . 





For example: 

y = -z — s — — , , ^, ■ gives for x = 2, -; hence, it 

^ 5 x^ — 21^7^+24^ — 4^ '0' ' 

is also admissable to put 

d(Sx^— Ix""— 8^ + 20) 9^^ — 14^— 8 

y = ~ 



d (b x''—2l x^ -^ 24:x— 4) 15^2 — 42 a; +24 
For X = 2, however, y is still = --; hence, we put again: 

— d( 9 x'—Ux— 8) _ 18 ^ — 14 _ 'd X— 1 _ 11 
^ ~ 8 (15 x^ — ^2 X -i- 24) ~ 30 ^ — 42 ~ 15 ^ — 21 "~ 9 * 
But the factor x — 2 is also in reality contained twice in the nu- 
merator and denominator of the given function. If both are divided 
by a; — 2, there results 

_ 3a;2 — ^ — 10 
^ ~ 5 ^2— 11 ^ 4- 2' 
and if this division be repeated in the last value : 

_ 3_^J-_5 
2^ — 5^ — 1' 

therefore, putting a? = 2, there foUows y = —, 

V 



Art. 36.] ELEMENTS OF ANALYSIS. 57 

_ ^, a — Va" — X . 

Further, y = gives for ^ = 0, -. 

CO V 

But we have 

e (a _ Va" — x) = — 'd{a' — xf = y^r^ 



X 
2 



1 



hence, there follows for this case, y = / 7'^_ — ^^- 

nat. log. x 

If, further, my = - /- , we put x = 1^ there follows V = q'j 

but we have now, 

d nat. log. x = — , and 8 1/ 1 — x = — — , . hence: 

X 2 1/1 — ^' 

2 V \ — x 2 . 



y = -- 



= -^- = Q. 



X 1 

Lastly, 

1 — sin. X + COS. X . TT 

y = ■ — rn • h gi^'®^ for a; = - (90"), 

-^ — 1 -^ sin. X + COS. X ^ . 2 ^ ^' 

_ 1 _ 1 -f _ 0^ 

y ~ _i + 1 + "" 0' 

hence, we must also put 

8 (1 — sin. X -f- (ios. x^ — COS. x — sin. x 



y 



8 ( — 1 -\- sin. X -\- COS. x) cos. x — sm. x 

— — 1 



— 1 



= 1. 



Art. 36. If, for a function y = au -{- /Sv^ a series of associated 
values of the variables u^ v, and y^ has been found by observation or 
measurement, we may also demand those values of the constants 
a and /? which are as free as possible from incidental and irregular 
errors of observation and measurement, and which, therefore, express 
as accurately as possible the connection between the magnitudes 
w, V, and 2/, of which u and v msiy also signify known functions of 
one and the same variable, x. Of all the rules which are employed 
in determining those values of the constants which are supposed to 
be the most accurate, the method of the least squares has the most 
general and scientific foundation. 

If 

' u,. v,, y^ 



1") ^1' 

%^ ^z-> y. 



U , V 



1 ^n5 



Vn 



58 ELEMENTS OF ANALYSIS. [Art. 36. 

are the observed results corresponding to the function y^=au-\- ftv, 
we have, for the errors of observation and their squares, the follow- 
ing values: 

^2 = 2/2 — (« '^^2 + /5 '^2) ) 



and 

r^^ = 2/f — 2rn(^ y^ — 2^v^ y^ + a^ ii} + 2a/?w^ v^ + /5^ v,^ 

'I = y'l — 2«i^2 y, — 2l3v.^ 2/2 + a' u.^ + 2 a/5 16.^ -u^ + /52 -u^' 

23' = yi — 2aW3 2/3 — 2iSu3 2/3 + «' '^3 + 2a/?W3 ■<;3 + /S^ v,' 



^n = 2/n — 2 a?i^ ?y^ _ 2,?'^;^ y^ + a^ ^^.^ + 2 aj3u^ v^ + /S^ v^ ^ 

and if, for abridgement, we employ the sign (-) of summation, in 

order to indicate a summation of homogeneous magnitudes, thus 

putting y^ + 2/./ + 2/3' H + !/b = - (2/0 ^ '"i 2/i + ^22/2 + ^^3 2/3 

+ • • • -{- v^y^ = 1' (^2/) 5 &c., we obtain for the sum of the squares 
of errors: 

I (z') = I (y') — 2aS (uy) — 2i3I (vy) -]- a"" I (u') 
+ 2a[3I (uv) + /522'(u2). 
In this equation, onlj^ the constants a and /? of the function y = 
au -j- /5u, which are to be regarded as independent, are unknown; 
excepting, of course, the sum of the squares of errors I (2:^, which 
must be treated as a dependent variable. The method of tlie least 
squares demands now that a as well as ft be so chosen that the sum 
of the squares 2* (z-) may become a minimum; and hence, we must 
differentiate the function obtained for w (2-) ; first, with respect to a, 
and again, with respect to /?, and put each of these differential quo- 
tients ^ 0. In this manner we arrive at the following equations 
of condition for a and /5: 

— I {uy) + al (u') -f ft I (uv) = 0, 

— y (vy) + ftl (d'-') + al (uv) = 0; 
the solution of which leads to the following expressions: 

I(v^)I(uy)-I(uv)I(vy) 



and 

2 (W) 2 {V') — 2 (UV) 2 [uv) 
S (u'-\ I" (vii\ — I (uv^ I (u.v^ 



2' (u') 2 (u'O — 2 (uv) 2 (uv) 
As we have here it = 1, therefore 2 (uv) = 2 (11), 2 (z^?/) = 2" (?/), 
and 2 (u-) ===: 1 -|- 1 -j- 1 -|- . . . = n, i. e. the number of the equa- 
tions or obserA^ations, the above formulae pass into the following: 



Art. 36.] 



ELEMENTS OF ANALYSIS. 



59 



/? 



^ (^') ^ (y) - ^ (^) - (^y) 

n I (vy) ~ Z (v) Z (y) 
n 1 (u-) — 1' (v) 1 (v)' 



i3v, where we have a = 0, 



For the still more simple fanction y 
there results 

" - I (0--) ' 
and lastly, for the simplest case, y = a^ where it is therefore requi- 
site to find the most probable value of one single magnitude, we have 

n 
as arithmetical mean of all the values found by measurement or 
observation. 

Example. — To become acquainted with the law of uniformly 
accelerated motion, i. e. with its initial velocity c and its measure 
of acceleration p, the spaces s^, s,^, s^, &c., corresponding to the 
different times ^^, ^^, t^^ &c., have been measured, and the following 
is the result. 



Times 


1 


o 
o 


5 


Y 


10 Sec. 


Spaces 





5 


20 


38 


581 


101 ft. 



If now s =■ ct -\- "^-^ be the law of this motion, we have to find the 

constants c and p. If, in the above formulae, we put u = t^ v = P^ 

= 0,/?=^, and y ^ s, there result the following formulae for 

the calculation of c and p-. 

_ !• (t*) I (St) - Z (f) I (st^ 
- J (P) X («') - 2- (f) 1' («») 

p y (f) s (sc) - z (t^) I (s() 

2 X (t-) X (<•) — X (f ) 2' («») ' 

whence, the following : 



t 


f 


f 


t' 


s 


St 


sf 


1 


1 


1 


1 


5 


5 


5 


3 


9 


2t 


81 


20 


60 


180 


5 


25 


125 


625 


38 


190 


950 


1 


49 


343 


2401 


58,5 


409,5 


2866,5 


10 


100 


1000 


10000 


101 


1010 


10100 


Amounts 


184 


1496 


13108 


222,5 


1674,5 


14101,5 




= ^ (n 


= ^ (n 


= ^ (^0 


= I(s) 


= I{st) 


= Z(sO. 



60 



ELEMENTS OF AJ^ALYSIS. 



[AuT 3t. 



From the above there results 
13108 . 1674,5 — 1496 



c = 



iP = 



14101,5 



184 . 13108 — 1496 
184 . 14101,5 — 1496 



1496 
J 6 74^ 

l496" 



85340 

17386 
89624 



184 . 13108 —1496 . 1496 173860 

and hence, the following formula for the observed motion: 

s = 4,908 t + 0,5155 . t\ 
According to this formula, we have 



= 4,908 ft. and 
= 0,5155 ft., 



for the times . . 





1 


3 


5 


7 


10 sec. 


the spaces . . . 





5,43 


19,36 


37,43 


59,62 


100,63 ft. 



Fig. 48. 




If the times (t) be taken as 
abscissas, and the observed, as 
well as the estimated, spaces 
(s) as ordinates, a curve AB, 
Fig.48, may be drawn through 
the ends of the estimated co- 
ordinates, which will pass be- 
tween the points M^N^O^P^Q 
determined by the observed 
co-ordinates, in such a manner 
that the sum of the squares 
of the deviation of the same 



from these points will be as 



Fig. 49. 



t = 1 

small as possible on both sides. 

Art. 37. If, in default of a formula for the constant progression 
of a magnitude y, or its dependence upon another magnitude ^, it 
be necessary to determine a value of the magnitude y which cor- 
responds to a given value of ^, by 
means of the values of x and y as 
known by experience or taken from 
a table, we must make use of the pro- 
cess of interpolation^ of which only 
the most important part is to be com- 
municated here. 

If the abscissas A jr = x. 



=: ,a?j, and A 31., = ^.,, Fig. 49, and 
the corresponding ordinates 3f^ P^ = 

Vo^ ^^1 A = 2/i, ^^^^t^ ^^2 A = y-i '^i^e 
C given, the ordinate 31 P = y corre- 
sponding to a new abscissa A3I = ^, may be expressed by the for- 
mula y = a -\- i3x -\- yoo^, provided the three points P^,P.^,P^ thus 




[Art 3T. ELEMENTS OF ANALYSIS. 61 

determined, lie in a nearly straight line, or in a slightly curved 
arc. If the initial point of the co-ordinates be transferred from A to 
Ji"^, the universality does not suffer, but we obtain simply y = a 
for j; = 0, and consequently, the constant member a = y^. 

If now we introduce into the assumed equation, first, x^ and y^^ 
and again, oc^ and y^^ we obtain the two following equations of con- 
dition: 

y^ — y^ = I3x^ -f- yx,^ and 

2/2 — 2/0 "^ /^^2 "i~ 7^2: from which there follows 

13 = (y.-yJ^{-('A-y.)^' and 

^ _ (V: — !/o) ^2 — (^2 — Vo) ^1 

/ 9 9 • 

1 O ** 1 



We have, consequently 



= y I r C^l — ^o) ^^' — (1/2 — Vo) ^' "^ 

, f ivi — y,) ^2 — (^2 — !/o) ■^ i'] 



If the ordinate y^ lay midway between y^ and 2/2, there would be 
x^ = 2x^^ and then, more simply, 

'3 !/o — 4 ^^ + ?/2-\ ^ , /-?/„ — 2 y^ -f 2/2 



2/ = 2/0 



(^^.f^O - + (""^^) 



X' 



If onlj^ two pairs of co-ordinates x^^ 3/^, and x^, ?/^, be given, we 
must regard the boundary line F^ P^ as a straight line, and conse- 
quently, put 

y =1 y^ -{- ftx^ as also 

2/1 = 2/0 + (^^i'> 



whence there results 



/? = ^^ ^ and 



2/ = 2/o + (^^)^. 

If it be required to interpolate b}^ construction a fourth ordinate 
y between the ordinates y^^ y^, y^^ we must describe a circle through 
the termini P^, P^, P^, of these ordinates, and take y equal to the 
ordinate of the same. The centre G of this circle is determined in 
the following well known manner, viz : let the points P,,, P^, Pg, be 
connected together by straight lines, and let perpendiculars be 
drawn to the centres of these lines ; the point G of intersection of 
these perpendiculars is the centre sought. 

If the distances of the point P, from the other points P^ and P^ 
be s„ and s^j and if the distance P^ K of the i3oint Pj from the con- 



62 ELEMENTS OF ANALYSIS. Art. 38. 

necting line s^ = P^ P^ be = Tt, there will result for the angle at 
circumference a = P^P^P^ = ^ the angle P^ G P^ at the centre : 



sin. a. 



h 



J 



and consequently, for the radius of gyration G P = G P^= CP^ 



= 0P,: 



r = 



2 sin. a 2h 

Therefore, we may find the centre G of the circle passing through 
Pq, Pj, P.^, if, with the radius from P^^, P^, or P^, calculated according 
to this formula, we intersect the perpendicular erected in the middle 
D of the chord P„ P^. 

Art. 38. The mean of the aggregate ordinates above the line of 
base M^ M^, is the height of a rectangle M^ M^ N^ N^ upon the same 
line of base, which has the same area as the surface M^ M^ P^ P^ P^, 
and may, therefore, be easily determined from this area. From 
Art. 29, we have the same: 






X 



IS.x^ , yoo^ 



3 

,, ^ _, [ (Vi — Vo) ^' — (y, — Vo) ^' 1 ^ 
^o-^^-r y ^^ ^2 _ ^^ ^. J 2 



, f (yi — y,) ^2 — (^2 — yd -^A ^ 
= C„ _L. (?/! — 2/o) ^' _ (!/2 — y.) (3 -^, — 2 ^,) ^ 

l^° "^ 6 ^-^ (^, — ^^) '6 (^2 — x;) J ^ 

-V 2 J^^ + l 6 x^ {x^ - x^) J"'^' 

and consequently, the mean ordinate, 

F _ y, +J/2 ,_ r (!/i — !/o) ^2 — (^2 — !/o) ^l ") ^ ^ 

^ 6 ^, r^„ — X,) J '^' 



y. 



^, 2 V. 6^,(^2 — ^,) 



If we had ^^ ~ = — , we should have to consider a rectilinear 

yi — yo^i 

boundary, and there would then be, simply, 

as also, 

^ __2V±J/2 

If, further, we had merely x^ = 2 x^, therefore, y^ equally distant 
from the boundary ordinates y^^ and y^^ there would be 



[Art. 38. 



ELEMENTS OF ANALYSIS. 



63 



cc„ 



■^= (!/o + 4 2/1 + yd -^- (vid. Art. 30), and 

„^ ^ 2/0 + 47/, + y. 

If an area M^M^P.^P^, Eig. 50, 
is determined by four co-ordinates 

M^ P„ = 2/0, ^^1 P^ = y,, ^. P. =2/., 
M^P^ = y^^ which are at equal dis- 
tances from each other, the mag- 
nitude of the same may be deter- 
mined approximately in the follow- 
ing manner. 

If we represent the line of base 
M^ Jig by ^3, and three ordinates 
N^Q^^ N^Q.^-, N.^Qsf inserted between 
y^ and y^ at equal distances from 
each qther, by z^^ z^, z,,, we can ]put 
the surface approximately; 

^o^s^a^o = -^= (i 2/0 + ^0 + ^x + ^2 + i y-s) ^~. 
But we have now, 

L 9,7. J^ 9,^ 9.7.-\-% 




Nj^Mi K^ M3N3 



^1 + ^2 + h _ 25J- 2 z^ -f- 2 ^3 _ 2 z^ + 2:, , 2 2:3 -f z^ ^^^ 



6 



6 



6 



2/1 = ^1 + -3- (^2 — ^1) = "^'3 ^ as also 2/, = -^^3-^ 

hence there follows ^» "^ !,^ "^ ^^ = ^^ "j" ^^ , and 

3 2 



V. 



= [^0 + 3 (y, + 2/2) + 2/3] -^5 as also 



Whilst the foregoing formula for y^ is applicable when the surface 
is resolved into an even number of strips, the latter is employed 
when the number of these portions is an odd one. 

Consequently, we may also put approximately: 

ydx =j if {x) dx = [2/0 -1- 3 (y, + 2/2) + 2/3] ~^^' 
if 

are four determined values of the function y = (f {x). 



64 ELEMENTS OF ANALYSIS. [Art. 38. 

(vid. example, Art. 30), we have c = 1, 

Cj = 2, and (x) = -; hence, there follows 

and the approximate value of this integral: 

J^2Q^ 111 

.— = [l + 3(H-|) + i]. 1=^ = 0,694. 



